Answer:
There are a total of [tex] { 6 \choose 3} = 20 [/tex] functions.
Step-by-step explanation:
In order to define an injective monotone function from [3] to [6] we need to select 3 different values fromm {1,2,3,4,5,6} and assign the smallest one of them to 1, asign the intermediate value to 2 and the largest value to 3. That way the function is monotone and it satisfies what the problem asks.
The total way of selecting injective monotone functions is, therefore, the total amount of ways to pick 3 elements from a set of 6. That number is the combinatorial number of 6 with 3, in other words
[tex] {6 \choose 3} = \frac{6!}{3!(6-3)!} = \frac{720}{6*6} = \frac{720}{36} = 20 [/tex]
16. In a group of 68 students, each student is registered for at least one of three classes – History, Math and English. Twenty-five students are registered for History, twenty-five students are registered for Math, and thirty-four students are registered for English. If only three students are registered for all three classes, how many students are registered for exactly two classes?
Answer:
Step-by-step explanation:
Total number of students = 68
Let history = H
Maths = M
English = E
n(H) = 25
n(M) = 25
n(E) = 34
n(HnMnE) = 3
Total = n(H) + n(M) + n(E) - people in exactly two groups + 2(people in exactly 3 groups) + people in none of the groups
68 = 25 + 25 + 34 - people in exactly two groups - 6 +0
68 = 84 -6 - people in exactly two groups
68 = 78 - people in exactly two groups
People in exactly two groups = 78 - 68
= 10
OR
From the venn diagram, people in exactly two groups are represented by x, y and z
Total = 25 - x - y - 3 + 25 - x - z - 3 + 34 - y - z - 3 + x + y + z + 3
68 = 50 - x - 3 + 34 - y - z - 3
68 = 84 - 6 - x - y - z
68 = 78 - x - y - z
68 - 78 = - x - y - z
-10 = -(x + y + z)
x+y+z = -10/-1
x+y+z = 10
The number of students that registered for exactly two courses = 10
Answer:
[tex[ two groups=-68+25+25+34 -6=10[/tex]
Step-by-step explanation:
For this case we have a diagram of the situation on the figure attached.
And for this case we can us the following rule from probability:
[tex]P(AUBUC) = P(A)+P(B) +P(C) -P(A and B)- P(A andC)-P(B and C) +P(A and B and C)[/tex]
Or equivalently:
[tex] total= A+ B +C - two groups -2*[people in3 groups]+ nonegroups[/tex]
We know the total on this case total=68, the people for history is A=25, for math B=25 and for english C=34, and the people in all the groups 3, so we can replace:
[tex]68=25+25+34- two groups -2*3 +0[/tex]
And if we solve for the poeple in two groups we got:
[tex[ two groups=-68+25+25+34 -6=10[/tex]
What is the probability of two consecutive ones in the string of 1s and 0s of length n?
The probability of getting two consecutive ones in a string of 1s and 0s of length n can be calculated as (n-1) / (2n).
Explanation:The probability of getting two consecutive ones in a string of 1s and 0s of length n can be calculated by considering the number of possible outcomes and the number of favorable outcomes.
Let's assume that the probability of getting a one is p and the probability of getting a zero is q. The favorable outcomes are when two consecutive ones occur, which can be represented as (1,1).
To calculate the probability, we need to determine the number of ways we can arrange the numbers in the string. Since we are only interested in the position of the ones, we can ignore the zeros.
Therefore, the number of favorable outcomes is n-1, because we have n-1 places where two consecutive ones can occur in a string of length n.
The number of possible outcomes is 2n, because each digit in the string can be either a one or a zero. Therefore, the probability of getting two consecutive ones is:
p = (n-1) / (2n)
I need help with this math question
Answer:
a.) y = 3/(2ˣ)
b.) y = 1/(2ˣ)
c.) y = (π^π)ˣ
d.) y = (1/27)(1/√(3))ˣ
e.) y = .002908/(.119025ˣ)
f.) y = .00000004808/(.0413ˣ)
Step-by-step explanation:
Concept need to know is:
a negative exponent will flip the numerator with the denominatora fraction as an exponent is just a root. so if the exponent is x^(1/2) then the root is 2 and x^(1/3) is a cube rootadding and subtracting exponent is the same thing as multiplying the same base. so x^(1+2) = (x^1)(x^2)and x^(1-2) = (x^1)(x^-2)1. Quadrilateral ABCD has vertices A(-1, 1), B(2, 3), C(6, 0) and D(3, -2). Determine using coordinate geometry whether or not the diagonals* of ABCD a. Bisect each other. B. Are congruent. C. Are perpendicular.1. Quadrilateral ABCD has vertices A(-1, 1), B(2, 3), C(6, 0) and D(3, -2). Determine using coordinate geometry whether or not the diagonals* of ABCD a. Bisect each other. B. Are congruent. C. Are perpendicular.
Answer:
The Conclusion is
Diagonals AC and BD,
a. Bisect each other
b. Not Congruent
c. Not Perpendicular
Step-by-step explanation:
Given:
[]ABCD is Quadrilateral having Vertices as
A(-1, 1),
B(2, 3),
C(6, 0) and
D(3, -2).
So the Diagonal are AC and BD
To Check
The diagonals AC and BD
a. Bisect each other. B. Are congruent. C. Are perpendicular.
Solution:
For a. Bisect each other
We will use Mid Point Formula,
If The mid point of diagonals AC and BD are Same Then
Diagonal, Bisect each other,
For mid point of AC
[tex]Mid\ point(AC)=(\dfrac{x_{1}+x_{2} }{2},\dfrac{y_{1}+y_{2} }{2})[/tex]
Substituting the coordinates of A and C we get
[tex]Mid\ point(AC)=(\dfrac{-1+6}{2},\dfrac{1+0}{2})=(\dfrac{5}{2},\dfrac{1}{2})[/tex]
Similarly, For mid point of BD
Substituting the coordinates of B and D we get
[tex]Mid\ point(BD)=(\dfrac{2+3}{2},\dfrac{3-2}{2})=(\dfrac{5}{2},\dfrac{1}{2})[/tex]
Therefore The Mid point of diagonals AC and BD are Same
Hence Diagonals,
a. Bisect each other
B. Are congruent
For Diagonals to be Congruent We use Distance Formula
For Diagonal AC
[tex]l(AC) = \sqrt{((x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2} )}[/tex]
Substituting A and C we get
[tex]l(AC) = \sqrt{((6-(-1))^{2}+(0-1)^{2} )}=\sqrt{(49+1)}=\sqrt{50}[/tex]
Similarly ,For Diagonal BD
Substituting Band D we get
[tex]l(BD) = \sqrt{((3-2))^{2}+(-2-3)^{2} )}=\sqrt{(1+25)}=\sqrt{26}[/tex]
Therefore Diagonals Not Congruent
For C. Are perpendicular.
For Diagonals to be perpendicular we need to have the Product of slopes must be - 1
For Slope we have
[tex]Slope(AC)=\dfrac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
Substituting A and C we get
[tex]Slope(AC)=\dfrac{0-1}{6--1}\\\\Slope(AC)=\dfrac{-1}{7}[/tex]
Similarly, for BD we have
[tex]Slope(BD)=\dfrac{-2-3}{3-2}\\\\Slope(BD)=\dfrac{-5}{1}[/tex]
The Product of slope is not -1
Hence Diagonals are Not Perpendicular.
Two cars enter the Delaware turnpike at shoreline drive at 8 am, each heading for ocean city. One cars average speed is 10 mph faster than the other. The faster car arrives at ocean city at 11 am, a half hour before the slower car. What is the average speed of each car?
Average speed of slower car = 60 mph
Average speed of faster car = 70 mph
Solution:
Given that Two cars enter the Delaware turnpike at shoreline drive at 8 am, each heading for ocean city
Given that One cars average speed is 10 mph faster than the other
Let "x" be the average speed of slower car
Then x + 10 is the average speed of faster car
The faster car arrives at ocean city at 11 am, a half hour before the slower car
Time taken by faster car:
The faster car arrives at ocean city at 11 am but given that they start at 8 a.m
So time taken by faster car = 11 am - 8am = 3 hours
Time taken by slower car:
The faster car arrives at ocean city at 11 am, a half hour before the slower car
So slower car takes half an hour more than faster car
Time taken by slower car = 3 hour + half an hour = [tex]3\frac{1}{2} \text{ hour}[/tex]
Now the distance between Delaware turnpike at shoreline drive and ocean city will be same for both cars
Let us equate the distance and find value of "x"
The distance is given by formula:
[tex]distance = speed \times time[/tex]
Distance covered by faster car:
[tex]distance = (x + 10) \times 3 = 3x + 30[/tex]
Distance covered by slower car:
[tex]distance = x \times 3\frac{1}{2} = x \times \frac{7}{2} = 3.5x[/tex]
Equating both the distance,
3x + 30 = 3.5x
3x - 3.5x = -30
-0.5x = -30
x = 60
Thus average speed of slower car = 60 mph
Average speed of faster car = x + 10 = 60 + 10 = 70 mph
Plz help, 15 points!
What is the slope of a line that is perpendicular to a line whose equation is 5y=10+2x
-------
Would it be -5/2 as in only the 5 is negative or the whole fraction is negative?
Answer: The slope of the line would be -5/2, and the whole fraction would be negative.
Step-by-step explanation:
Technically just the 5 is negative because the new slope is going down 5, yet on paper its just safer to write the whole fraction as a negative.
Two buses use the stop at Meadowland Road and Mockingbird Lane. Bus 143 stops every 30 minutes on its express route to downtown. Bus 62 stops there every 8 minutes on its west side route. If they were both at the stop at 6:30 AM, when will they arrive together again?
Answer:
8:30 AM
Step-by-step explanation:
Given:
Bus 143 stops every 30 minutes on its express route to downtown
and Bus 62 stops there every 8 minutes on its west side route
both meet at the bus stop at 6.30 AM
to find when they would meet the second time we need to find LCM( 30,8)
= 120
Hence the two buses would meet after 120 minutes that is 2 hours.
Hence they would be meet at 6.30+2= 8.30 AM
For a parade, a large spherical balloon is filled with a gas that weighs 200 grams per cubic meter. When filled with the gas, the balloon has a diameter of 9 meters. What is the total weight, in grams, of the gas in a filled balloon? Use 3.14 for π .
Answer:
76302 grams
Step-by-step explanation:
Diameter of the balloon = 9 m
Density of the gas (weight occupied per m³) = 200 g per m³
Volume of the spherical balloon = [tex]\frac{4}{3}\pi r^{3}[/tex]
Here r = 4.5 m
Then V= [tex]\frac{4}{3}\times\ 3.14 \times (4.5)^{3}[/tex]
V = 381.51 m³
Total weight = Total volume occupied * Density
= 381.51*200 = 76302 grams.
Final answer:
The total weight of the gas in a filled balloon with a 9-meter diameter is 76,340.25 grams, calculated by first finding the volume of the balloon and then multiplying it by the weight of the gas per cubic meter.
Explanation:
To find the total weight of the gas in a filled spherical balloon, first, we need to find the volume of the balloon using the formula for the volume of a sphere, which is V = (4/3)πr³. The diameter of the balloon is given as 9 meters, so the radius r is 9/2 meters or 4.5 meters.
Calculating the volume of the balloon:
V = (4/3)π(4.5 m)³ = (4/3)π(4.5³) = (4/3) × 3.14 × (91.125) ≈ 381.70125 cubic metersNow, we'll multiply the volume of the balloon by the weight per cubic meter of the gas to get the total weight of the gas:
Total weight = Volume × Weight per cubic meter = 381.70125 m³ × 200 g/m³ = 76340.25 gramsif f is a differentiable function and f(0)=-1 and f(4)-3 then which of the following must be true there exists a c in [0,4] where f(c)=0
Answer:
True, see proof below.
Step-by-step explanation:
Remember two theorems about continuity:
If f is differentiable at the point p, then f is continuous at p. This also applies to intervals instead of points.(Bolzano) If f is continuous in an interval [a,b] and there exists x,y∈[a,b] such that f(x)<0<f(y), then there exists some c∈[a,b] such that f(c)=0.If f is differentiable in [0,4], then f is continuous in [0,4] (by 1). Now, f(0)=-1<0 and f(4)=3>0. Thus, we have the inequality f(0)<0<f(4). By Bolzano's theorem, there exists some c∈[0,4] such that f(c)=0.
The question is about the intermediate value theorem in calculus, which states a function takes on every value between f(a) and f(b) in a closed interval [a, b]. Given the function's values at 0 and 4, it must cross the x-axis, or equal zero, somewhere in the interval [0, 4].
Explanation:This is a question about intermediate value theorem, a fundamental theorem in calculus. The intermediate value theorem states that if a function is continuous on a closed interval [a, b], then it takes on every value between f(a) and f(b) at some point within that interval. If we know that f(0) = -1 and f(4) = 3, this means the function f crosses the x-axis (or, in other words, f(c) = 0 for some c) somewhere in the interval [0, 4] because zero lies between -1 and 3 (the function's values at 0 and 4 respectively).
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Which of the following must also be known in order to compute the standard deviation?A) ModeB) MeanC) RangeD) Median
Answer:
B) Mean
Step-by-step explanation:
Standard Deviation is the measure of the amount of variation or dispersion of a set of values. Standard Deviation is represented by lower case Greek alphabet sigma σ. Standard Deviation is the square root of variance. Variance is the average of the squared differences or variation or dispersion from the Mean. Therefore, to compute standard deviation, the mean of the given data must be known.
Standard Deviation, σ = [tex]\sqrt{Variance}[/tex]
σ = [tex]\sqrt({\frac{1}{N-1}}[/tex] ∑[tex]_{i = 1}^{N}(x_{i} - X)^{2})[/tex]
where
[tex]x_{1},x_{2},x_{3}, ...x_{N},[/tex] are the values of the sample observed,
X is the mean value of these observations, and
N is the number of observations in the sample.
How much of the sphere do you have to paint red so that no matter ow you put a cube in it, some corner touches red?
Explanation:
To achive that you have to piant a ring in red, which will have its big diameter equal to the sphere diameter and its small diameter equal to the cube diagonal.
The diagonal of the cube can be calculated using Pithagoras:
[tex]D^2=L^2+L^2[/tex]
Where D is the diagonal and L is the side of the cube
The perimeter of a rectangle is 360m. If it's length is decreased by 20% and it's breadth is increased by 25%, we get the same perimeter. Find the dimensions of the rectangle.
Answer:the length of the rectangle is 100m
the width of the rectangle is 80m
Step-by-step explanation:
Let L represent the length of the rectangle.
Let W represent the width of the rectangle.
The formula for determining the perimeter of a rectangle is expressed as
Perimeter = 2(L + B)
The perimeter of a rectangle is 360m. This means that
2(L + B) = 360
L + B = 180 - - - - - - - - - -1
If it's length is decreased by 20%, its new length would be
L - 0.2L = 0.8L
it's breadth is increased by 25%, it means that the new breadth
B + 0.25B = 1.25B
Since the perimeter remains the same,
2(0.8L + 1.25B) = 360
0.8L + 1.25B = 180 - - - - - - - - -2
Substituting L = 180 - B into equation 2, it becomes
0.8(180 - B) + 1.25B = 180
144 - 0.8B + 1.25B = 180
- 0.8B + 1.25B = 180 - 144
0.45B = 36
B = 36/0.45 = 80
L = 180 - B = 180 - 80
L = 100
Which of the following functions is graphed below?
The function graphed below is
y = x³ - 3, x ≤ 2
= x² + 6, x > 2 ⇒ C
Step-by-step explanation:
The graph has two parts:
The first part start at x > 2 and its a part of parabola (quadratic function)The second part End at x = 2 and its a part of a cubic function∴ The graph represents two functions:
→ Quadratic function y = ax² + bx + c with domain x > 2
→ Cubic function y = ax³ + bx² + cx + d with domain x ≤ 2
∵ The answers have two functions:
→ y = x³ - 3 ⇒ cubic function
→ y = x² + 6 ⇒ quadratic function
∵ The domain of the cubic function is x ≤ 2
∵ The domain of the quadratic function is x > 2
∴ The answer is
y = x³ - 3, x ≤ 2
x² + 6, x > 2
Learn more:
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Answer:
y={x^3-3, x ≤ 2
{x^2+6, x>2
Step-by-step explanation:
a pex :)
You interview four people. You find that person A is a married 24-year-old white female with 2 pets and whose college GPA was 2.4. Person B is a single 44-year-old Asian female with 5 pets and whose college GPA was 2.5. Person C is a single 35-year-old Hispanic male with 4 pets and whose college GPA was 2.2. Person D is a single 48-year-old Asian male with 3 pets and whose college GPA was 3.8. Construct a data file for the characteristics number of pets and ethnic group for this sample.
Answer:
The data file is included in attached excel file.
Step-by-step explanation:
There are seven variables in the data file for four persons interviewed. Person type, marital status, ethnic group and gender are categorical variables while age, number of pets and GPA are quantitative variables. Person type is classified as A,B,C and D. Marital status consists of category married and single. Age of persons lies between 20-50. Ethic group has three categories that are white, Asian and Hispanic. Gender of persons has two categories male and female. Number of pets for the persons interviewed lies in the range of 2 to 5. Last GPA variable ranges from 2.2 to 3.8.
In a certain large city, 45% of families earn less than $35,000 per year. Assuming the distribution is binomial and you can use the exact binomial calculation, what's the probability, accurate to the number of decimal places given, that a simple random sample of 30 families will have 10 or fewer families earning less than $35,000 per year? Use the exact binomial calculation.
Answer:
The probability is %13,5
Step-by-step explanation:
If it is a binomial distribution function than we can find a probability of earning less or more than 35000$ in this certain large city. Lets assume that p is probability of earning less than 35000$ and q is earning more than 35000$.:
p=0,45
q=0,55
So general formula of n families that earning less than 35000$ is:
[tex]P(X=n)=combination(30,n)*0,45^n*0,55^{30-n}[/tex]
Probability of 10 or less families out of 30 families that earning less than 35000$ is:
[tex]combination(30,10)*0.45^{10}*0.55^{20}+combination(30,9)*0.45^9*0.55^{21}+combination(30,8)*0.45^8*0.55^{22}+combination(30,7)*0.45^7*0.55^{23}+combination(30,6)*0.45^6*0.55^{24}+combination(30,5)*0.45^5*0.55^{25}+combination(30,4)*0.45^4*0.55^{26}+combination(30,3)*0.45^3*0.55^{27}+combination(30,2)*0.45^2*0.55^{28}+combination(30,1)*0.45^{1}*0.55^{29}+combination(30,0)*0.45^0*0.55^{30}=0,135[/tex]
The correct probability that a simple random sample of 30 families will have 10 or fewer families earning less than $35,000 per year, using the exact binomial calculation, is approximately 0.0674.
To solve this problem, we will use the binomial probability formula, which is given by:
[tex]\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]
Given that 45% of families earn less than $35,000 per year, we have p = 0.45.
The number of families in the sample is n = 30
We want to find the probability that 10 or fewer families earn less than $35,000 per year, so k will range from 0 to 10.
Then, we sum these probabilities to find the cumulative probability of 10 or fewer families earning less than $35,000 per year.
Using the binomial probability formula, we calculate:
[tex]\[ P(X \leq 10) = \sum_{k=0}^{10} \binom{30}{k} (0.45)^k (0.55)^{30-k} \][/tex]
This calculation can be done using statistical software or a calculator that supports binomial probability calculations.
After performing the exact binomial calculation, we find that the probability is approximately 0.0674.
This is the probability that, in a random sample of 30 families from this city, 10 or fewer families will earn less than $35,000 per year.
Suppose that the current equilibrium price of silver is $34 per ounce. If silver is produced under conditions of perfect competition and the industry is in long-run equilibrium, the average total cost of producing silver: a. is less than $34 per ounce. b. is $34 per ounce. c. is indeterminate. d. exceeds $34 per ounce.
Answer:
b. is $34 per ounce
Step-by-step explanation:
If the production cost were less, a competitor would drive the price down. If the production cost were more, the supplier would go out of business.
Since we're at equilibrium, the production cost must be equal to $34 per ounce.
Which of the following ordered pair could NOT be included in this set if it is a function? { (-4, 2), (3, 6), (4, 3), (x, y) }
(x, y) = (4, -3)
(x, y) = (3, 6)
(x, y) = (1, 2)
(x, y) = (-3, 6)
Why is this the answer?
The x value x = 4 shows up in the point (4, 3), which is in the given function set. Adding (4, -3) to this set will have x = 4 show up twice. We cannot have one x value pair up with more than one y value. In other words, any input cannot map to more than one output. Visually, the two points (4,3) and (4,-3) will fail the vertical line test, which means we wouldnt have a function.
State the vertical asymptote of the rational function. F(x)=(x-6)(x+6)/x^2-9.
X=6,x=-6
X=3,x=-3
X=-6,x=6
None
Answer:
x = 3, x = -3
Step-by-step explanation:
The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the values that x cannot be and if the numerator is non zero for these values then they are vertical asymptotes.
Solve : x² - 9 = 0 ⇒ x² = 9 ⇒ x = ± 3
The vertical asymptotes are x = - 3 and x = 3
The vertical asymptotes of the rational function F(x) = (x - 6)(x + 6)/(x^2 - 9) are x = -3 and x = 3.
Explanation:The rational function F(x) = (x - 6)(x + 6)/(x^2 - 9) has vertical asymptotes at x = -3 and x = 3.
To find the vertical asymptotes, we need to determine the values of x that make the denominator equal to zero. In this case, the denominator is x^2 - 9, which can be factored as (x - 3)(x + 3).
Therefore, the vertical asymptotes occur at x = -3 and x = 3, since these are the values that make the denominator zero.
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During a football game a concert in stenciled family three hamburgers and two white dogs for a total of $13 and another family two hamburgers and five hotdogs for a $ $16 what are the prices of a hamburger and a hotdog
Answer:
Price of hamburgers = $2
Price of white-dogs = $3
Step-by-step explanation:
Let
The number price of hamburgers be x
The number price of white-dogs be y
Then the first family buys 3 hamburgers and 2 white -dogs for $13
3x + 2y = 13-----------------------------(1)
Another first family buys 2 hamburgers and 5 white -dogs for $16
2x + 5y = 16----------------------------(2)
To solve let us multiply eq(1) by 2 and eq(2) by 3, we get
6x + 4y = 26-----------------------------(3)
6x + 15y = 48----------------------------(4)
subracting (3) from (4)
6x + 15y = 48
6x + 4y = 26
(-)
-------------------------
11y = 22
---------------------------
[tex]y =\frac{22}{11}[/tex]
y =2
Now substituting the value of y in eq(1),
3x + 2(2) = 13
3x + 4 = 13
3x = 13-4
3x = 9
x =[tex]\frac{9}{3}[/tex]
x =3
2. A savings account is started with an initial deposit of $600. The account earns 2.1 % interest compounded annually.
(a) Write an equation to represent the amount of money in the account as a function of time in years.
(b) Find the amount of time it takes for the account balance to reach $800. Show your work.
Answer:
Step-by-step explanation:
Initial amount that was deposited into the savings account is $600 This means that the principal,
P = 600
The account earns 2.1 % interest compounded annually.. This means that it was compounded once in a year. So
n = 1
The rate at which the principal was compounded is 2.1%. So
r = 2.1/100 = 0.021
It was compounded for t years. So
t = t
a) The formula for compound interest is
A = P(1+r/n)^nt
A = total amount in the account at the end of t years. Therefore
A = 600 (1+0.021/1)^1×t
A = 600(1.021)^t
b)when A =$800, it becomes
800 = 600(1.021)^t
Dividing both sides by 600, it becomes
1.33 = (1.021)^t
Taking the tth root of both sides
t = 14 years
It will take 14 years
What is the gross profit? A. $130,000 B. $120,000 C. $230,000 D. $300,000 Previous Next
Answer: check explanation
Step-by-step explanation:
The question is not complete but nonetheless, gross profit is easy to calculate. Let us start by the definition of gross profit, to how to calculate gross profit and, calculation of gross profit margin.
GROSS PROFIT: gross profit can be defined as the profit gained after subtracting the costs of making and selling the products. Gross profit is also known as gross margin.
HOW TO CALCULATE GROSS PROFIT: Gross profit can be calculated by subtracting the total revenue from the cost of goods sold. That is, gross profit= total revenue - cost of goods sold.
So, for example; Company A makes women handbags. Assuming the company made $10million in total revenue for the year and cost of goods sold is $5 million. We can use the formula above to find the company's gross profit margin.
Hence, company A's gross profit= total revenue($10,000,000) - cost of goods sold ($5,000,000).
= $5,000,000.
That is, the gross profit for company A= $5,000,000.
GROSS PROFIT MARGIN: This can be calculated using the formula below;
Gross profit margin= (total revenue - cost of goods)/ revenue.
Hence, from the example above;
Gross profit margin= $10,000,000 - $5,000,000) / $10,000,000.
= 50%
How would you use a completely randomized experiment in each of the following settings? Is a placebo being used or not? Be specific and give details.
(a) A veterinarian wants to test a strain of antibiotic on calves to determine their resistance to common infection. In a pasture are 22 newborn calves. There is enough vaccine for 10 calves. However, blood tests to determine resistance to infection can be done on all calves.
A- Use random selection to pick 5 calves to inoculate.
B- No placebo is being used.
C- After inoculation, test 5 calves to see if there is a difference in resistance to infection between the two groups.
D- After inoculation, test 10 calves to see if there is a difference in resistance to infection between the two groups.Use random selection to pick 10 calves to inoculate.
E- A placebo is used for the remaining 12 calves.
F- After inoculation, test all calves to see if there is a difference in resistance to infection between the two groups.
(b) The Denver Police Department wants to improve its image with teenagers. A uniformed officer is sent to a school one day a week for 10 weeks. Each day the officer visits with students, eats lunch with students, attends pep rallies, and so on. There are 18 schools, but the police department can visit only half of these schools this semester. A survey regarding how teenagers view police is sent to all 18 schools at the end of the semester. (Select all that apply.)
A- After the police visits, survey all the schools to see if there is a difference in views between the two groups.
B- After the police visits, survey 9 of the schools to see if there is a difference in views between the two groups.
C- No placebo is being used.
D- A placebo is used for the remaining 9 schools.
E- Use random selection to pick nine schools to visit.
F- Use random selection to pick 18 schools to visit.
(c) A skin patch contains a new drug to help people quit smoking. A group of 75 cigarette smokers have volunteered as subjects to test the new skin patch. For one month, 40 of the volunteers receive skin patches with the new drug. The other volunteers receive skin patches with no drugs. At the end of the two months, each subject is surveyed regarding his or her current smoking habits.
A- Then record the smoking habits of the 35 volunteers to see if a difference exists between the two groups.
B- No placebo is being used.
C- Use random selection to pick 35 volunteers for the skin patch with the drug.
D- Then record the smoking habits of the 40 volunteers to see if a difference exists between the two groups.
E- Use random selection to pick 40 volunteers for the skin patch with the drug.
F- A placebo patch is used for the remaining 35 volunteers in the second group.
G- Then record the smoking habits of all volunteers to see if a difference exists between the two groups.
Answer:
(a)
Use random selection to pick 10 calves to inoculate.(you didn't mention this option in the question instead you mix it with other option)
(B).No placebo is being used.
(F).After inoculation, test all calves to see if there is a difference in resistance to infection between the two groups.
(b)
(C).No placebo is being used.
(E).Use random selection to pick nine schools to visit.
(A).After the police visits, survey all the schools to see if there is a difference in views between the two groups.
(c)
(F.)A placebo patch is used for the remaining 35 volunteers in the second group.
(E).Use random selection to pick 40 volunteers for the skin patch with the drug.
(G).Then record the smoking habits of all volunteers to see if a difference exists between the two groups.
Step-by-step explanation:
In question one the answer is mixed with other option so the correct answer statements are provided along with the option for all questions. Placebo is not used in question (a) scenario because it is mentioned in the statement that there is enough vaccine is present and sample of 10 calves is selected from 22 calves and we test all calves after the inoculation because it is mention that blood test should be done on all calves.
Placebo is not used in question (b) scenario because it is mentioned in the statement that the police department can visit half of 18 schools and random sample of 9 schools are selected and it is mentioned in the statement that survey is sent to all 18 school so after the police visit all schools are surveyed to see the difference in views
Placebo patch is used in question (c) because it is mentioned in the statement that 40 out of 75 received the new skin patch so, the remaining 35 received placebo patch and random sample of 40 volunteers are selected for new skin patch. It is mentioned in the statement that each subject is survey so in the end smoking habits of all volunteers are recorded.
In the given settings, a completely randomized experiment can be used to test the effects of different interventions. Placebo is not being used in these experiments.
Explanation:In a completely randomized experiment, the researcher randomly assigns subjects to different treatment groups to determine the effect of the treatment. In the given settings:
(a) A veterinarian wants to test a strain of antibiotic on calves. A completely randomized experiment can be conducted by randomly selecting 5 calves from the pasture to inoculate with the antibiotic while the remaining 17 serve as the control group with no placebo being used. After inoculation, the resistance to infection can be tested on both groups to compare the difference.
(b) The Denver Police Department wants to improve its image with teenagers. A completely randomized experiment can be conducted by randomly selecting half of the 18 schools to receive the visits from the police officer while the other half serve as the control group with no placebo being used. Then, a survey can be sent to all 18 schools at the end of the semester to compare the views between the two groups.
(c) A skin patch contains a new drug to help people quit smoking. A completely randomized experiment can be conducted by randomly selecting 40 volunteers to receive the skin patch with the drug while the remaining 35 volunteers receive the placebo patch with no drugs. The smoking habits of both groups can be surveyed at the end of the two months to see if a difference exists between the two groups.
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Let w be the amount Juan earned walking dogs then (2w-6) is the amount Juan earned mowing lawns the total amount Juan earned is w +(2w-6) or w-6 Emma wrote the expression 2(3w-6) to represent the amount of the money that Juan spent is she correct explain
Answer:
The expression written by Emma to represent the amount of the money that Juan earned is WRONG.
Step-by-step explanation:
Here, the amount earned by Juan by walking dogs = w
The amount earned byJuan lowing lawns = (2w - 6)
Now, the total amount earned by Juan
= Amount earned from (Mowing Lawn + Walking dogs)
= w + (2 w - 6)
On further simplification, we get:
w + (2 w - 6) = w + 2w - 6 = 3w-6 = 3(w-2)
So, the total amount earned by Juan = 3(w-2)
Now, the expression written by Emma = 2(3w-6)
and 2(3w-6) ≠ 3(w-2)
Hence, the expression written by Emma to represent the amount of the money that Juan earned is WRONG.
70 sixth-graders had an average weight of 90 pounds. 30 seventh graders had an average weight of 100 pounds. What was the average weight of all these students?
We know that 70 sixth graders had an average weight of 90 pounds, and 30 seventh graders had an average weight of 100 pounds.
We know that the weight of all 70 sixth graders will be 70*90, since we're multiplying the total amount of students by the average.
Using this same logic tells us that the total weight for the 30 seventh graders is 30*100
The average weight for all 100 students will be the total weight of the sixth graders (which is 70*90) added with the total weight of the seventh graders (which is 30*100), then finally divided by the total amount of students in the sample space (which is 100).
[tex]\dfrac{70(90)+30(100)}{100}[/tex]
[tex]=\dfrac{6300+3000}{100}[/tex]
[tex]=\dfrac{9300}{100}[/tex]
[tex]=93[/tex]
The average of all 100 students is 93 pounds.
Let me know if you need any clarifications, thanks!
To fulfill the requirements for a certain degree, a student can choose to take any 7 out of a list of 20 courses, with the constraint that at least 1 of the 7 courses must be astatistics course. Suppose that 5 of the 20 courses are statistics courses.(a) How many choices are there for which 7 courses to take?(b) Explain intuitively why the answer to (a) is not
Complete answer:
Fulfill the requirements for a certain degree, a student can choose to take any 7 out of a list of 20 courses, with the constraint that at least 1 of the 7 courses must be a statistics course. Suppose that 5 of the 20 courses are statistics courses.
(a) How many choices are there for which 7 courses to take?
(b) Explain intuitively why the answer to (a) is not [tex] \binom{5}{1}\binom{19}{6} [/tex]
Answer:
a) 71085 choices
b) See below
Step-by-step explanation:
a) First we're going to calculate in how many ways you can take 7 courses from a list of 20 without the constraint that at least 1 of the 7 courses must be a statistics course, that's simply a combination of elements without repetition so it's: [tex]\binom{20}{7} [/tex], but now we should subtract from that all the possibilities when none of the courses chose are a statistic course, that's is [tex]\binom{15}{7} [/tex] because 15 courses are not statistics and 7 are the ways to arrange them. So finally, the choices for which 7 courses to take with the constraint that at least 1 of the 7 courses must be a statistics course are:
[tex]\binom{20}{7}-\binom{15}{7}=71085 [/tex]
b) It's important to note that the constraint at least 1 of the 7 courses must be a statistics course make the possible events dependent, we can not only fix an statistic course and choose the others willingly ( that is what [tex] \binom{5}{1}\binom{19}{6} [/tex] means) because the selection of one course affect the other choices.
In this exercise, we have to use our knowledge of statistics to calculate how many options can be chosen for a course, so we find that:
a) 169 choices
b) 1 of the 7 courses must be a statistics course, because the selection of one course affect the other choices.
So from the data reported in the exercise we can say that:
a) First we're going to calculate in how many ways you can take 7 courses from a list of 20 without the constraint that at least 1 of the 7 courses must be a statistics course, that's simply a combination of elements without repetition so it's. So we have that;
[tex]C=\frac{m!}{p!(m-p)!}\\C=\frac{20!}{7!(20-7)!}\\C=169[/tex]
b) It's influential to note that the restraint not completely 1 of the 7 courses must be a enumeration course form the attainable occurrence determined by, we can not only fix a statistic course and select the possible choice gladly because the preference from among choices of individual course influence the added selection.
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Josh has a rope that is 43 feet long. He cuts off a piece that is 10 feet long and gives it to his brother. He wants to cut the rest of the rope into pieces measuring exactly 5 feet each. How many pieces can he cut?
Answer:
6
Step-by-step explanation:
Answer:
Step-by-step explanation:
The initial length of Josh's rope was 43 feet. He cuts off a piece that is 10 feet long and gives it to his brother. This means that the length of the remaining rope would be
43 - 10 = 33 feet.
He wants to cut the rest of the rope into pieces measuring exactly 5 feet each. To determine the number of pieces that he can cut, it becomes
33/5 = 6.6
Therefore, only 6 pieces measuring exactly 5 feet each can be cut.
Analytic function on unit disk with power series has pole on unit circle, then power series diverges on unit circle.
Answer:
The function
{\ displaystyle f (z) = {\ frac {z} {1- | z | ^ {2}}}} {\ displaystyle f (z) = {\ frac {z} {1- | z | 2}
It is an example of real and bijective analytical function from the open drive disk to the Euclidean plane, its inverse is also an analytical function. Considered as a real two-dimensional analytical variety, the open drive disk is therefore isomorphic to the complete plane. In particular, the open drive disk is homeomorphic to the complete plan.
However, there is no bijective compliant application between the drive disk and the plane. Considered as the Riemann surface, the drive disk is therefore different from the complex plane.
There are bijective conforming applications between the open disk drive and the upper semiplane and therefore determined as Riemann surfaces, are isomorphic (in fact "biholomorphic" or "conformingly equivalent"). Much more in general, Riemann's theorem on applications states that the entire open set and simply connection of the complex plane that is different from the whole complex plane admits a bijective compliant application with the open drive disk. A bijective compliant application between the drive disk and the upper half plane is the Möbius transformation:
{\ displaystyle g (z) = i {\ frac {1 + z} {1-z}}} {\ displaystyle g (z) = i {\ frac {1 + z} {1-z}}}
which is the inverse of the transformation of Cayley.
if an analytic function on the unit disk has a pole on the unit circle, its power series representation diverges on the unit circle, as the singularity prevents the power series from converging outside the disk of convergence.
To understand why a power series diverges on the unit circle when the analytic function it represents has a pole on the unit circle, we can use the concept of analytic continuation and the properties of poles and singularities.
Here's a step-by-step explanation:
1.Analytic function on the unit disk: Let's consider an analytic function defined on the open unit disk, denoted by[tex]\(D = \{z \in \mathbb{C} : |z| < 1\}\)[/tex]. This means the function is holomorphic (complex differentiable) at every point within this disk.
2.Power series representation: Since the function is analytic on[tex]\(D\)[/tex], it can be represented by a power series expansion around any point [tex]\(z_0\) in \(D\)[/tex]. Let's denote this function by[tex]\(f(z)\)[/tex], and its power series representation centered at[tex]\(z_0\) by \(\sum_{n=0}^{\infty} a_n (z - z_0)^n\)[/tex].
3.Pole on the unit circle: Suppose[tex]\(f(z)\)[/tex] has a pole (a point where the function becomes unbounded) on the unit circle[tex]\(|z| = 1\)[/tex], i.e., there exists a point [tex]\(z_1\)[/tex] on the unit circle such that [tex]\(f(z_1)\)[/tex] is infinite. Without loss of generality, let's assume [tex]\(z_1 = 1\)[/tex] (since the unit circle is symmetric about the origin).
4.Behavior near the pole: Near the pole at [tex]\(z = 1\)[/tex], the function[tex]\(f(z)\)[/tex]can be expanded in a Laurent series, which includes negative powers of [tex]\((z - 1)\)[/tex]. This expansion will have infinitely many terms with negative powers, indicating the singularity at [tex]\(z = 1\)[/tex].
5.Radius of convergence: The radius of convergence of the power series representation of [tex]\(f(z)\)[/tex]is at least the distance from the center of convergence to the nearest singularity. In this case, since the singularity (pole) is on the unit circle, the radius of convergence of the power series cannot exceed 1.
6.Divergence on the unit circle: Since the radius of convergence of the power series representation of [tex]\(f(z)\)[/tex] is at most 1, the power series diverges at every point on the unit circle (except possibly at the point of singularity itself, where it may converge by definition). This divergence occurs because the function has a singularity (pole) on the unit circle.
Therefore, if an analytic function on the unit disk has a pole on the unit circle, its power series representation diverges on the unit circle, as the singularity prevents the power series from converging outside the disk of convergence.
If $-5\leq a \leq -1$ and $1 \leq b \leq 3$, what is the least possible value of $\displaystyle\left(\frac{1}{a} \frac{1}{b}\right)\left(\frac{1}{b}-\frac{1}{a}\right) $
Answer:
The least possible value is -2, obtained for a = -1 and b = 1.
Step-by-step explanation:
We want the minimum of [tex] \displaystyle\left(\frac{1}{a} \frac{1}{b}\right)\left(\frac{1}{b}-\frac{1}{a}\right) [/tex] , where
[tex] -5 \leq a \leq -1 [/tex][tex] 1 \leq b \leq 3 [/tex]First, lets simplify the expression given, we use common denominator on the second part, using ab as the common denominator. We obtain
[tex] \frac{1}{b} - \frac{1}{a} = \frac{a-b}{ab} [/tex]
As a result
[tex] \displaystyle\left(\frac{1}{a} \frac{1}{b}\right)\left(\frac{1}{b}-\frac{1}{a}\right) = \frac{a-b}{(ab)^2} [/tex]
we need the minimum of the function
[tex] f(a,b) = \frac{a-b}{(ab)^2} [/tex]
with the restrictions [tex] -5 \leq a \ -1, 1 \leq b \leq 3 [/tex]
First, we calculate the gradient of f and find where it takes the zero value.
[tex]\nabla{f} = (f_a,f_b) [/tex]
with
[tex]f_a = \frac{(ab)^2 - (a-b) 2ab^2}{(ab)^2} = -1 + 2 \frac{b}{a}[/tex]
Since it has the reversed sign, we get
[tex]f_b = - (-1 + 2 \frac{a}{b}) =1 - 2 \frac{a}{b}[/tex]
In order for [tex] \nabla{f} [/tex] to be zero, we need both [tex] f_a [/tex] and [tex] f_b [/tex] to be zero, observe that
[tex]f_b = 0 \, \rightarrow 1 -2\frac{a}{b} = 0 \, \rightarrow 1 = 2 \frac{a}{b} \, \rightarrow b = 2a [/tex]
Which is impossible with the given restrictions. Hence, the minimum is realized in the border.
If we fix a value a₀ for a, with a₀ between -5 and -1 the function g(b) = f(a₀,b) wont have a minimum for b in [1,3] because the partial derivate of f over b didnt reach the value 0 in the restrictions given. On the other hand. by making a similar computation that before, we can obtain that the partial derivate of f over the variable a doesnt reach the value 0 either. This means that f doesnt reach the minimum on the sides. As a consecuence, it reach a minimum on the corners.
The 4 possible corner values are (-5,1), (-5,3), (-1,1) and (-1,3)
[tex] f(-5,1) = \frac{-6}{25} = -0.24 [/tex]
[tex] f(-5,3) = \frac{-8}{225} = -0.0355 [/tex]
[tex] f(-1,1) = \frac{-2}{1} = -2 [/tex]
[tex] f(-1,3) = \frac{-4}{9} = -0.444 [/tex]
Clearly the least possible value between the four corners is -2.
The table shows the amount of money in your savings account over a period of 6 weeks. You plan to keep saving at the same rate until you have $825 in the account. Which equation could you use to find the number of weeks,n, it would take to reach your goal?
A 55⋅n=825
B 55⋅n+100=825
C 100−n=825
D 100⋅n+55=825
Answer:
Choice B
Step-by-step explanation:
Looks like interest isn't being used, so that makes it simple.
Every week the same amount is added, that means we can use a linear equation, which is what it is asking us to pick between so it all works out, awesome.
Now, we need two things to make an equation. the slope and two points. Definitely have points so we need the slope.
The slope is found by taking any two points and finding the difference of their y values and dividing that by the distance of their x values. so find two points (x1, y1) and (x2, y2) and then use the formula(y2-y1)/(x2-x1) Also, we will say number of weeks is the x and the amount of money is y
No matter which point you use you will get the slope is 55
From there we find the function with the formula y - y1 = m(x - x1)
We know m and I reccomend using (0, 100) as our x1 y1 because 0 will usually make things easier.
y - y1 = m(x - x1)
y - 100 = 55(x-0)
y = 55x + 100
you could have used any point again, 0 just means it goes away if you add 0 or subtract 0. Of course your problem uses n so just replace x with n. and y is the value we want to end at, which is 825
825 = 55n + 100, or arranging it so it's the same as the option, 55*n + 100 = 825, so that's choice B
The equation that calculates the weeks it would take to reach the target is : 55n = 825.
What is a mathematical function, equation and expression?function : In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.expression : A mathematical expression is made up of terms (constants and variables) separated by mathematical operators.equation : A mathematical equation is used to equate two expressions.Given is the table shows the amount of money in your savings account over a period of 6 weeks and you plan to keep saving at the same rate until you have $825 in the account.
Assume that it would take [n] weeks to reach the target. The unit rate from the given table can be calculated as -
m = (155 - 100)/(1 - 0)
m = 55
So, we can write the equation as -
55n = 825
Therefore, the equation that calculates the weeks it would take to reach the target is : 55n = 825.
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A balloon is rising vertically above a level, straight road at a constant rate of 1 ft/sec. Just when the balloon is 65 feet above the ground, a bicycle passes under it, going 17 ft/sec. How fast is the distance between and bicycle and the balloon changing 3 seconds later?
To calculate the rate at which the distance between the bicycle and the balloon is changing, take into account the speeds of the balloon and bicycle and use Pythagoras's theorem and the formula dd/dt = sqrt((dx/dt)^2 + (dy/dt)^2). This calculates out to approximately 17.03 ft/sec.
Explanation:This problem can be solved using the concepts of relative velocity and Pythagoras' theorem. It's a classic problem of a balloon moving vertically upwards at a speed of 1 ft/sec and a bicycle moving horizontally at a speed of 17 ft/sec.
After 3 seconds, the balloon would have ascended an additional 3 feet (because it's rising at 1 foot per second), so it will be at a height of 68 feet. Meanwhile, in those same 3 seconds, the bicycle will have travelled 51 feet (because it's going at 17 feet per second).
At this point, you can use Pythagoras's theorem to find the distance (d) between the bicycle and the balloon. The distance d is the hypotenuse of a right triangle where one side (the vertical side) is 68 feet and the other side (the horizontal side) is 51 feet. Using Pythagoras' theorem, d² = 68² + 51². So, d = sqrt(68^2+51^2) = 85 feet.
Then, find the change in d, known as dd/dt, which tells us how fast the distance between the balloon and the bicycle is changing. Given the constant velocities of the balloon and the bicycle, dd/dt is also a constant. You can use the formula dd/dt = sqrt((dx/dt)^2 + (dy/dt)^2), where dx/dt = 17 ft/sec (the speed of the bicycle), and dy/dt = 1 ft/sec (the speed of the balloon). So, dd/dt = sqrt(1^2 + 17^2) = sqrt(290) = approximately 17.03 ft/sec. So, the distance between the balloon and the bicycle is increasing at a rate of 17.03 ft/sec after 3 seconds.
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The distance between the bicycle and the balloon is changing at a rate of approximately 11 ft/sec, 3 seconds after the bicycle has passed under the balloon.
We need to determine how fast the distance between the bicycle and the balloon is changing 3 seconds after the bicycle passes under the balloon.
Let's denote:
y as the height of the balloon above the groundx as the horizontal distance between the bicycle and the point directly beneath the balloonz as the distance between the bicycle and the balloonWe are given:
Balloon’s rise rate: 1 ft/secBicycle’s speed: 17 ft/secInitial height of the balloon: 65 feetAfter 3 seconds:
Balloon’s height: y = 65 + 3 × 1 = 68 ftHorizontal distance travelled by bicycle: x = 3 × 17 = 51 ftApply the Pythagorean theorem to find z:
z = √(x² + y²)
z = √(51² + 68²)
z = √(2601 + 4624)
z = √7225
z = 85 ft
To find the rate of change of the distance z, we differentiate with respect to time (t):
2z (dz/dt) = 2x (dx/dt) + 2y (dy/dt)
Continue with substituting the known values:
85 (dz/dt) = 51 (17) + 68 (1)
85 (dz/dt) = 867 + 68
dz/dt = 935 / 85
dz/dt ≈ 11 ft/sec
Thus, the distance between the bicycle and the balloon is changing at a rate of approximately 11 ft/sec after 3 seconds.