Final answer:
When a positive integer x, which leaves a remainder of 5 when divided by 9, is multiplied by 3, the remainder when 3x is divided by 9 is 6.
Explanation:
When the positive integer x is divided by 9, the remainder is 5. To determine the remainder when 3x is divided by 9, we should remember that when two positive numbers multiply, the result has a positive sign. So multiplying x by 3, we get 3x. The initial information tells us that x = 9n + 5 for some integer n. Thus, 3x equals 3(9n + 5), which simplifies to 27n + 15. This expression can be further broken down as 27n + 9 + 6, which is the same as 9(3n + 1) + 6. Therefore, when 3x is divided by 9, it is the same as dividing this expression by 9, which leaves us with a remainder of 6. Hence, the remainder when 3x is divided by 9 is 6.
A drug is eliminated from the body through urine. Suppose that for a dose of 10 milligrams, the amount A(t) remaining in the body t hours later is given by A(t) = 10(0.7)t and that in order for the drug to be effective, at least 4 milligrams must be in the body.
a. Determine when 2 milligrams is left in the body.
b. What is the half-life of the drug?
Answer: a - 4.512 hours
b - 1.94 hours
Step-by-step explanation:
Given,
a) A(t) = 10 (0.7)^t
To determine when 2mg is left in the body
We would have,
A(t) = 2, therefore
2 = 10(0.7)^t
0.7^t =2÷10
0.7^t = 0.2
Take the log of both sides,
Log (0.7)^t = log 0.2
t log 0.7 = log 0.2
t = log 0.2/ 0.7
t = 4.512 hours
Thus it will take 4.512 hours for 2mg to be left in the body.
b) Half life
Let A(t) = 1/2 A(0)
Thus,
1/2 A(0) = A(0)0.7^t
Divide both sides by A(0)
1/2 = 0.7^t
0.7^t = 0.5
Take log of both sides
Log 0.7^t = log 0.5
t log 0.7 = log 0.5
t = log 0.5/log 0.7
t = 1.94 hours
Therefore, the half life of the drug is 1.94 hours
If a rectangular region has perimeter P inches and area A square inches, is the region square?(1) P = 4/3*A(2) P = 4âA
Answer:
(1) Insufficient data
(2) Sufficient data
Step-by-step explanation:
We need to check whether the given data is sufficient or not to prove that a rectangular region has perimeter P inches and area A square inches, is the region square.
Assume that the given conditions are
1. [tex]P=\frac{4}{3}A[/tex]
2. [tex]P=4\sqrt{A}[/tex]
Area of square is
[tex]A=a^2[/tex]
Taking square root on both sides.
[tex]\sqrt{A}=a[/tex] .... (1)
where a is side length.
Perimeter of square is
[tex]P=4a[/tex] ... (2)
From (1) and (2) we get
[tex]P=4\sqrt{A}[/tex]
It means second condition is sufficient to prove that the rectangle reason is square, because it is true for all.
For a=3,
A=9 and P=12
[tex]12=\frac{4}{3}(9)[/tex]
[tex]12=12[/tex]
LHS=RHS
For a=6,
A=36 and P=24
[tex]24=48[/tex] (False statement)
The first condition is insufficient because it may or may not be true.
A woman standing on a cliff is watching a motor boat though a telescope as the boat approaches the shoreline directly below her. If the telescope is 250 feet above the water and if the boat is approaching at 20 feet per second, at what rate is the angle of the telescope changing when the boat is 250 feet from shore
Answer:
Dα/dt = 0.079 degree/sec
Step-by-step explanation:
From problem statement, is easy to see, that if point A is ubicated at the top of the telescope, the shoreline is directly below the woman ( point B), and the point where the boat is, which is at distance x from shoreline is point C. These three point shape a right triangle with angle α (the angle of the telescope).
So we have
tan α = x/250
Differentiating both sides of the equation we get
D (tan α)/dt = ( 1/250)* Dx/dt
sec² α Dα/dt = ( 1/250)* Dx/dt
we already know that Dx/dt = 20 feet/sec
sec² α Dα/dt = 20/250 ⇒ sec² α Dα/dt = 0.08
Dα/dt = 0.08 / sec² α
Then
tan α = 20/250 = 0,08 α = arctan 0.08 α ≈ 5⁰
Dα/dt = 0.08/ sec² α
From tables we get cos 5⁰ = 0.9961 then
1/ 0.9961 = 1.003
sec α = 1.003 and sec² α = 1.0078
Dα/dt = 0.08/ sec² α ⇒ Dα/dt = 0.08/1.0078
Dα/dt = 0.079 degree/sec
The change of position of the boat results in a change of the angle of the telescope, which can be calculated using related rates. With known factors such as the speed of the boat and its distance from shore, the rate of change of the telescope's angle can be found using the principles of trigonometry and calculus.
Explanation:This question is about related rates in calculus. Related rates describe the relationship between different rates of change that are connected to each other. In this case, the changing position of the boat creates a change in the angle of the telescope.
We know the woman is watching a boat that is approaching a shoreline directly below her at 20 feet per second, and we are asked to find the rate that the angle of the telescope is changing when the boat is 250 feet from shore. Here is a way of visualizing it:
Let D be the distance of the boat from the base of the cliff and θ the angle that the telescope makes with the horizontal. We are given that D(t) decreases at 20 feet per second and that when D(t) = 250, we want to know what is dθ/dt.
By using trigonometry, we can find a relationship between D and θ. Specifically, tan(θ) = 250/D, so by implicit differentiation, (sec^2(θ)) * dθ/dt = -250/D^2 * dD/dt. From the given data, dD/dt = -20 and D = 250, so substitute them into the equation and evaluate θ using the tan–1(1) to obtain dθ/dt.
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A rectangular poster is 3 times as long as it is wide. A rectangular banner is 5 times a long as it is wide. Both the banner and the poster have perimeters of 24 inches. What are the lengths and widths of the poster and banner?
Answer:
The answer to your question is
a) Poster width = 3 in
lenght = 9 in
b) banner
width = 2 in
lenght = 10 in
Step-by-step explanation:
a) For the poster
width = x
lenght = 3x
Perimeter of a rectangle = 2base + 2 height
= 2(x) + 2(3x) = 24
2x + 6x = 24
8x = 24
x = 24/8
x = 3
width = 3 in
lenght = 3(3) = 9 in
b) For the banner
width = y
lenght = 5y
Perimeter of a banner = 2 base + 2 height
2 (y) + 2(5y) = 24
2y + 10y = 24
12y = 24
y = 24/12
y = 2
width = 2 in
lenght = 5(2) = 10 in
Suppose that 3 balls are chosen without replacement from an urn consisting of 3 white and 7 red balls. Assume moreover that the white balls are labeled 1; 2; 3. Let Xi = 1 if the i-th white ball is chosen among the 3 selected balls, and 0 otherwise. Find the pmf of (X1;X2).
Answer:
P(X1=1, X2=1) = 1/15
P(X1=1, X2=0) = 7/30
P(X1=0, X2=1) = 7/30
P(X1=0, X2=0) = 7/15
Step-by-step explanation:
Let Xi = 1 if the i-th white ball is selected. In this question the 3 white balls are marked 1,2 and 3.
We need to know the possible combination between X1 and X2 i.e. for the white ball 1 and 2 being chosen in the event.
We also need to note that the event is dependent which that after a ball is being chosen, it will not be put back hence affecting the probability of picking the next ball.
Consider all the possible combination between X1 and X2
a) both being chosen P(X1=1, X2=1)
= (3/10) x (2/9) = 1/15
Note that the first probability is the probability before any ball is being picked. The chances for ball white to be pick is 3/10 (3 white ball from the total 10 balls).
After 1 white ball being selected, that ball is not again out back into the urn making white ball 2 and total ball 9. Hence probability of picking another white ball is 2/9
b) only X1 chosen P(X1=1, X2=0)
= (3/10) x (7/9) = 7/30
After the white ball was picked, the probability of white not being pick again is the same as red being picked. Since there is still 7red balls and a total of 9 balls, the probability is 7/9
c) only X2 chosen P(X1=0, X2=1)
= (7/10) x (3/9) = 7/30
The white is not being picked first, making the probability of picking red is 7/10. Then the probability of white being picked is 3/9
d) both not chosen
P(X1=0, X2=0)
= (7/10) x (6/9) = 7/15
In other word only red being chosen. So the first probability is 7 red out of 10 balls (7/10), and the next red ball being picked next is 6/9
2sin^2x + 2cos^2x = 4a, then a = ?
A. 4
B. 3
C. 2
D. 1
E. 1/2
Answer:
E. 1/2
Step-by-step explanation:
Divide by 2, then make use of the Pythagorean identity for sine and cosine.
sin(x)^2 +cos(x)^2 = 2a
1 = 2a . . . . . . . sin²+cos²=1
1/2 = a
Answer:
Option E) is correct.
[tex]a=\frac{1}{2}[/tex]
Step-by-step explanation:
Given trignometric equation is [tex]2sin^{2}x + 2cos^{2}x = 4a[/tex]
To find the value of "a" from the given equation:
[tex]2sin^{2}x + 2cos^{2}x = 4a[/tex]
Taking common number "2" outside the equation of left hand side
[tex]2(sin^{2}x +cos^{2}x) = 4a[/tex]
[tex]sin^{2}x +cos^{2}x =\frac{4a}{2}[/tex]
[tex]sin^{2}x+cos^{2}x =2a[/tex]
( We know the trignometric formula [tex]sin^{2}\theta +cos^{2}\theta=1[/tex] here
[tex]\theta=x[/tex] )
Therefore [tex](1) =2a[/tex]
[tex]\frac{1}{2} =a[/tex]
It can be written as
[tex]a=\frac{1}{2}[/tex]
Therefore [tex]a=\frac{1}{2}[/tex]
Option E) is correct.
If a borrower's monthly interest payment on an interest-only loan at an annual interest rate of 7.3% is $877, how much was the loan amount (rounded to the nearest hundred)?A. $120,100B. $100,300C. $144,200D. $134,200
Answer:
C. $144,200
Step-by-step explanation:
We have been given that a borrower's monthly interest payment on an interest-only loan at an annual interest rate of 7.3% is $877.
To find the loan amount, we will use simple interest formula.
[tex]I=Prt[/tex], where,
I = Amount of interest,
P = Principal amount,
r = Annual interest rate in decimal form,
t = Time in years.
One month will be equal to 1/12 year.
[tex]7.3\%=\frac{7.3}{100}=0.073[/tex]
Upon substituting our given values in simple interest formula, we will get:
[tex]877=P*0.073*\frac{1}{12}[/tex]
[tex]877*12=P*0.073*\frac{1}{12}*12[/tex]
[tex]10524=P*0.073[/tex]
[tex]P*0.073=10524[/tex]
[tex]\frac{P*0.073}{0.073}=\frac{10524}{0.073}[/tex]
[tex]P=144164.3835616438356[/tex]
Upon rounding to nearest hundred, we will get:
[tex]P\approx 144,200[/tex]
Therefore, the loan amount was $144,200 and option C is the correct choice.
Final answer:
To calculate the loan amount based on a monthly interest payment and an annual interest rate, convert the rate to monthly and divide the payment by this rate. For a monthly payment of $877 at 7.3% annual interest, the loan amount is approximately $144,200.
Explanation:
To determine the loan amount of an interest-only loan given a monthly interest payment and an annual interest rate, follow these steps:
Convert the annual interest rate to a monthly rate by dividing by 12.Divide the monthly interest payment by the monthly interest rate to find the loan principal.In this case, the annual interest rate is 7.3%, so the monthly interest rate is 7.3% / 12 = 0.6083%. The monthly interest payment is $877. Therefore, the loan amount can be calculated as follows:
$877 / 0.6083% = $877 / 0.006083 = $144,200.46
When rounded to the nearest hundred, the loan amount is $144,200.
The correct answer is C. $144,200.
Mrs Cain coleslaw recipe calls for 1/3 cup of oil, 1/2 cup of vinegar, 1/4 cup of sugar. If she has 1 cup of vinegar, how much oil and sugar does she need to make 1 batch
Answer:
Step-by-step explanation:
Mrs Cain coleslaw recipe calls for 1/3 cup of oil, 1/2 cup of vinegar, 1/4 cup of sugar. This means that the ratio of the number of cups of oil to vinegar to sugar is 1/3 : 1/2: 1/4
If she has 1 cup of vinegar, the ratio that would correspond to the given ratio would be determined by multiplying the given ratio by 2. It becomes, 2/3 : 2/2 : 2/4 = 2/3 : 1 : 1/2
Therefore, she would require 2/3 cups if oil and 1/2 cups of sugar to make 1 batch.
PLEASE HELP
which graph represents the given inequality y < 3x-4
Answer: Here is the answer.
Step-by-step explanation:
If natalie and her friends decide to rent 4 lanes at reguler cost for a party ten people need to rent shoes and 4 people are members what is the total cost for the party
The question is missing a tabular data. So, it is attached below.
Answer:
The total cost for the party is $74.50.
Step-by-step explanation:
Given:
Lanes rented at regular cost = 4
Cost of 1 lane rented at regular cost = $9.75
Cost of 1 lane rented for members = $7.50
Cost of 1 shoe rental at regular cost(non members) = $3.95
Cost of 1 shoe rental for members = $2.95
Since, lanes are rented at regular cost, we use unit rate at regular cost
So, cost of 4 lanes rented = [tex]4\times 9.75= \$ 39[/tex]
Now, out of 10 people who rented shoes, 4 are members. So, the number of non-members is given as:
Non members who rented shoes = 10 - 4 = 6
So, 4 members and 6 nonmembers rented shoes.
So, cost of 6 non members renting shoes = [tex]6\times 3.95=\$ 23.70[/tex]
Cost of 4 members renting shoes = [tex]4\times 2.95=\$ 11.80[/tex]
Total cost for the party is the sum of all the costs. This gives,
= 39 + 11.80 + 23.70
= 50.80 + 23.70
= $74.50
Therefore, the total cost for the party is $74.50.
A residual plot has data points that are all very close to the x-axis. What does this say about the data?
A) The line of best fit will be a horizontal line.
B) A linear model is appropriate.
C) There is not enough information to determine this.
D) A non-linear model is appropriate.
Answer:
B
Step-by-step explanation:
because, the closer the data points are to the x axis on a residual plot with no definite shape means that a linear model is appropriate for this data set
A linear model is appropriate because, the closer the data points are to the x axis on a residual plot with no definite shape. So, option B is correct.
What is residual plot?A residual is a measure of how far away a point is vertically from the regression line.
It is the error between a predicted value and the observed actual value.
A residual plot is a graph that has data points that are all very close to the x-axis. It shows the residuals on the y axis and the independent variable on the x axis.
The goodness of fit of a linear model is depicted by the pattern of the graph of a residual plot. If each individual residual is independent of each other, they create a random pattern together.
When graphing the residual values you know if a linear model is an appropriate model for your data if the points in the residual plot are scattered.
A linear model is appropriate because, the closer the data points are to the x axis on a residual plot with no definite shape.
So, option B is correct.
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Alexander and Jo live 5 miles apart. They decide to leave their homes at 3 p.M. And travel toward each other. If Alexander bikes 14 miles per hour and Jo jogs 6 miles per hour, when will they meet?
Answer:
Step-by-step explanation:
Here is an illustration of the problem:
----------------------------->|<------------------
A t J
Alex and Jo start from their separate homes and drive towards one another. The t indicates the time at which they meet, which is the same time for both. Filling in a d = rt table:
d = r x t
Alex 14 t
Jo 6 t
The formula for motion is d = rt, so that means that Alex's distance is 14t and Jo's distance is 6t.
14t 6t
---------------------------------->|<------------------
A t J
The distance between them is 5 miles, so that means that Alex's distance plus Jo's distance equals 5 miles. In equation form:
14t + 6t = 5 and
20t = 5 so
t = .25 hours or 15 minutes.
If they leave their homes at 3 and they meet 15 minutes later, then they meet at 3:15.
Simplify the cubed root of six over the fourth root of six
six raised to the one twelfth power
six raised to the one fourth power
six raised to the four thirds power
six raised to the seven twelfths power
Answer:
six raised to the one twelfth power
Step-by-step explanation:
The cubed root of 6/the fourth root of 6 equals (6^1/3)/(6^1/4)
6^((1/3)-(1/4))
6^((4-3)/12)
6^1/12
The simplified form of the expression is six raised to the one twelfth power
Given the expression
[tex]\dfrac{\sqrt[3]{6} }{\sqrt[4]{6} }[/tex]According to indices, this expression can also be written as:
[tex]\dfrac{(6)^{1/3}}{6^{1/4}}[/tex]Using the law of indices;
[tex]\dfrac{a^m}{a^n} = a^{m-n}[/tex]
Applying this expression will give:
[tex]=\dfrac{6^{1/3}}{6^{1/4}} \\= 6^{1/3-1/4}\\=6^{4-3/12}\\=6^{1/12}[/tex]
Hence the simplified form of the expression is six raised to the one twelfth power
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On each round, Ann and Bob each simultaneously toss a fair coin. Let Xn be the number of heads tossed in the 2n flips which occur during the first n rounds. For each integer m > 0, let rm denote the probability that there exists an n such that Xn = m.
Answer:
[tex]P=\frac{2n!}{m!*(2n-m)!}*0.5^{2n}[/tex]
Step-by-step explanation:
In a coin toss the probability of tossing a head is 0.5 (50% head/50% tails)
If n is the number of rounds and 2n the number of coins tossed (one for each player), the probability of having m heads tossed is:
[tex]R=\frac{2n!}{m!*(2n-m)!}[/tex]
R is the number of cases (combination of coins tossed) that gives a m number of heads. Each case has a probability of [tex]P_{case}=0.5^{2n}[/tex] so:
[tex]P=\frac{2n!}{m!*(2n-m)!}*0.5^{2n}[/tex]
For example, to toss 4 heads in 5 rounds:
n=52n=10m=4[tex]P=\frac{10!}{4!*(10-4)!}*0.5^{10}[/tex]
[tex]P=\frac{10*9*8*7*6!}{4!*6!}*0.5^{10}[/tex]
[tex]P=\frac{10*9*8*7}{4!}*0.5^{10}[/tex]
[tex]P=\frac{10*9*8*7}{4!}*0.5^{10}=0.205[/tex]
An equation for the line graphed is
A) y =
3
2
x + 3
B) y =
1
2
x + 3
C) y = -
3
2
x + 3
D) y = -
1
2
x + 3
Answer:
The answer to your question is letter A
Step-by-step explanation:
Process
1.- Find two points of the line
A ( -2, 0)
B ( 0, 3)
2.- Find the slope
[tex]m = \frac{y2 - y1}{x2 - x1}[/tex]
[tex]m = \frac{3 - 0}{0 + 2}[/tex]
[tex]m = \frac{3}{2}[/tex]
3.- Find the equation of the line
y - y1 = m(x - x1)
y - 0 = [tex]\frac{3}{2} (x + 2)[/tex]
y = [tex]\frac{3}{2} (x + 2)[/tex]
Simplify
y = [tex]\frac{3}{2} x + 3[/tex]
A test car went 64 miles on 2 gallons in the morning, 16 miles on 1/2 gallon at noon, and 32 miles on 1 gallon at 5 PM.Is the relationship a proportional relationship? Explain.
It is proportional because the relationship between gallons of gas and distance is constant and linear. The rate is 32 miles per gallon. You can also call 32 the slope of this linear relationship.
Find the remainder when the polynomial $x^5 x^4 x^3 x^2 x$ is divided by $x^3-4x$.
Answer:
Remainder would be [tex]5x^2+21[/tex]
Step-by-step explanation:
Given,
Dividend = [tex]x^5+x^4+x^3+x^2+x[/tex]
Divisor = [tex]x^3-4x[/tex]
Using long division ( shown below ),
We get,
[tex]\frac{x^5+x^4+x^3+x^2+x}{x^3-4x}=x^2+x+5+\frac{5x^2+21}{x^3-4x}[/tex]
Therefore,
Remainder would be [tex]5x^2+21[/tex]
The local bike shop sells a bike and accessories package for $276. If the bike is worth 5 times more than the accessories, how much does the bike cost?
The cost of the bike for the given condition will be $230.
How to form an equation?Determine the known quantities and designate the unknown quantity as a variable while trying to set up or construct a linear equation to fit a real-world application.
In other words, an equation is a set of variables that are constrained through a situation or case.
Suppose the cost of one bike is x while accessories for one bike is y.
As per the given,
x + y = 276
The cost of bike is 5 times more then accessories,
x = 5y
5y + y = 276
6y = 276
y = 46
Bike cost = 5(46) = $230
Hence "The bike will cost $230 in the specified condition".
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Solve the system of equations: (Find the ordered pair that is a solution to both equations). y=4x-3y=-2x+9
Answer:
(-2,-11)
Step-by-step explanation:
x = -2
y = -11
Tim Has 480 Pokemon cards in his collection, which he arranges in his album. Each page of his album holds 12 cards. The album has 35 pages. Write an expression that shows how many Pokemon cards Tim Has left after he has filled his album completely?
Answer:
Step-by-step explanation:
Total number of Pokemon cards that Tim has in his collection is 480
He arranges the cards in his album and each page of his album holds 12 cards.
The total number of pages in the album is 35. After he has filled his album completely, the total number of cards that the album would contain would be 35 × 12 = 420
Let x represent the number of Pokemon cards he has left. Therefore, the expression becomes
x + 420 = 480
x = 480 - 420 = 60
James and Terry open a savings account that has a 2.75% annual interest rate, compounded monthly. They deposit $500 into the account each month. How much will be in the account after 20 years? A. $159,744.59 B. $48,407.45 C. $330,600.15 D. $580,894.18
Answer:
Option A is the answer(here the answer is calculated taking the whole value, without approximating it to a nearest value)Step-by-step explanation:
Annual interest rate is 2.75%. Hence, the monthly interest rate is [tex]\frac{2.75}{12}[/tex]
The amount will be compounded [tex](20\times12) = 240[/tex] times.
Every month they deposits $500.
In the first month that deposited $500 will be compounded 240 times.
It will be [tex]500\times [1 + \frac{2.75}{1200} ]^{240}[/tex]
In the second month $500 will be deposited again, this time it will be compounded 239 times.
It will give [tex]500\times [1 + \frac{2.75}{1200} ]^{239}[/tex]
Hence, the total after 20 years will be [tex]500\times [1 + \frac{2.75}{1200} ]^{240} + 500\times [1 + \frac{2.75}{1200} ]^{239} + ........+ 500\times [1 + \frac{2.75}{1200} ]^{1} = 160110.6741[/tex]
The account will have approximately $159,744.59 after 20 years.
Explanation:To calculate the future value of the savings account after 20 years with a 2.75% annual interest rate compounded monthly, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A is the future value of the accountP is the monthly deposit amount ($500)r is the annual interest rate (2.75% or 0.0275)n is the number of times the interest is compounded per year (12 for monthly compounding)t is the number of years (20)Plugging in the values, we can calculate:
A = 500(1 + 0.0275/12)^(12*20)
A = 500(1.00229167)^(240)
A ≈ $159,744.59
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When 25 students went to Washington dc 18 of them visited the museum of natural history and 22 visited the natural air and space museum. How many students visited both museums.
Answer:
15 students visited both museums.
Step-by-step explanation:
Operations With Sets
Sets are a collection of elements. Some sets have elements in common with other sets. These elements are said to be in their intersection. If we know the number of elements in the set A and in the set B, and also the total number of elements in both sets, we can say
[tex]N(A\bigcup B)=N(A)+N(B)-N(A\bigcap B)[/tex]
where [tex]N(A\bigcup B)[/tex] is the total number of elements, N(A) and N(B) are the number of elements in A and B respectively, and [tex]N(A\bigcap B)[/tex] is the number of elements in their intersection. If we wanted to know that last number, then we isolate it
[tex]N(A\bigcap B)=N(A)+N(B)-N(A\bigcup B)[/tex]
Let A= Students who visited the museum of natural history
B=Students who visited the natural air and space museum
We know [tex]N(A)=18, N(B)=22, N(A\bigcup B)=25,\ so[/tex]
[tex]N(A\bigcap B)=18+22-25=15[/tex]
Answer: 15 students visited both museums.
Note: We are assuming no students didn't visit at least one museum
The set theory in Mathematics implies that 15 out of the 25 students must have visited both the Museum of Natural History and Natural Air and Space Museum, since these students are counted twice when you add the students who visited each museum separately.
Explanation:The problem asked is: When 25 students went to Washington DC, 18 of them visited the Museum of Natural History and 22 visited the Natural Air and Space Museum. How many students visited both museums?
To answer this, we need to understand the concept of the set theory in Mathematics. It helps us know the total number of students who visited both museums: we subtract the total number of students from the sum of students who visited each museum separately.
So, if we add those who visited the Museum of Natural History (18 students) and those who went to the Natural Air and Space Museum (22 students), we get 40. But we know there were only 25 students in total, implying that 15 students must have visited both museums since these students are counted twice in our sum. Therefore, 15 students visited both museums.
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Find angle A in the following triangle.
A. 35.54
B. 35.67
C. 36.24
D. 36.77
Tan(Angle) = Opposite leg / Adjacent leg
Tan(A) = 5/7
A = Arctan(5/7)
A = 35.54 degrees.
Answer:
A. 35.54
Step-by-step explanation:
To find the angle A in the diagram above, we will simply use angle formula, we will check and see which angle formula will be best fit to use.
The formulas are;
SOH CAH TOA
sin Ф = opposite / hypotenuse
cos Ф = adjacent / hypotenuse
tan Ф = opposite / adjacent
Looking at the figure given, since we are to find angle A, then our adjacent is 7 and the opposite is 5.
since we are given both opposite and adjacent, then the best formula to use for this is tan Ф
tan Ф = opposite / adjacent
opposite = 5 and adjacent =7
we will go ahead and insert our values
tan A = 5/7
tan A = 0.71429
To get the value of tan A, we will simply take the [tex]tan^{-1}[/tex] of both-side
[tex]tan^{-1}[/tex] tan A = [tex]tan^{-1}[/tex] 0.71429
A = 35.54
Therefore angle A is 35.54
Let n be the smallest positive integer that is a multiple of 75 and has exactly 75 positive integral divisors, including itself and 1. Find n/75.
Answer:
[tex]n=2^4 3^4 5^2 =32400[/tex] and then we have:
[tex]\frac{n}{75}=\frac{2^4 3^4 5^2}{3 5^2}=432[/tex]
Step-by-step explanation:
From the info given by the problem we need an integer defined as the smallest positive integer that is a multiple of 75 and have 75 positive integral divisors, and we are assuming that 1 is one possible divisor.
Th first step is find the prime factorization for the number 75 and we see that
[tex]75=3 5^2[/tex]
And we know that 3 =2+1 and 5=3+2 and if we replace we got:
[tex] 75 = (2+1)(4+1)^2 = (2+1)(4+1)(4+1)[/tex]
And in order to find 75 integral divisors we need to satisify this condition:
[tex]n= a^{r_1 -1}_1 a^{r_2 -1}_2 *......[/tex] such that [tex]a_1 *a_2*....=75[/tex]
For this case we have two prime factors important 3 and 5. And if we want to minimize n we can use a prime factor like 2. The least common denominator between 2 and 4 is LCM(2,4) =4. So then the need to have the prime factors 2 and 3 elevated at 4 in order to satisfy the condition required, and since 5 is the highest value we need to put the same exponent.
And then the value for n would be given by:
[tex]n=2^4 3^4 5^2 =32400[/tex] and then we have:
[tex]\frac{n}{75}=\frac{2^4 3^4 5^2}{3 5^2}=432[/tex]
The smallest positive integer that is a multiple of 75 is 32400 and
integral divisors are 432.
Positive integer
Positive integers are the whole number that is greater than zero and do not include decimal or fraction values.
Find the smallest positive integer that is a multiple of 75 that has exactly 75 positive integral divisors.
How to calculate?We know that
[tex]75 = 3*5^{2}[/tex]
So the value of n which has 75 divisors then the multiplication of power of prime factor should be 75. The formula is given by
[tex]n = 2^{x-1} *3^{y-1} *5^{z-1}[/tex]
then the multiplication of x, y, and z must be 75.
x, y, and z are 5, 5, and 3 will be the values.
[tex]n = 2^{4} *3^{4} *5^{2} = 32400[/tex]
And it is divisible by 75 also.
[tex]\dfrac{n}{3*5^{2} } = \dfrac{2^{4} *3^{4} *5^{2}}{3*5^{2} } = 432[/tex]
Thus, the smallest positive integer that is a multiple of 75 is 32400 and integral divisors are 432.
More about the Integer link is given below.
https://brainly.com/question/1768254
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Find the area of shaded region to the nearest tenth.
804.2 yd2
603.2 yd2
201.1 yd2
1895.0 yd2
the answer of this question is 603.2yd2
Answer:Area of the shaded region is 603.2 yards^2
Step-by-step explanation:
The diagram contains two circles. The smaller circle has a radius of 8 yards.
The bigger circle has a radius of 16 yards.
The area of a circle is expressed as
Area of circle = πr^2
Where
r = radius of circle
π is a constant whose value is 3.142
The area of the smaller circle would be
3.142 × 8^2 = 201.088 yards^2
The area of the bigger circle would be
3.14 × 16^2 = 804.352 yards^2
Area of the shaded region would be area of the bigger circle - area of the smaller circle. It becomes
804.352 - 201.088 = 603.2 yards^2
A new crew of painters can paint a small apartment in 12 hours and experience crew can paint a small apartment in six hours how many hours does it take to paint the apartment on the true cruise work together
Answer:
The new crew and the experience crew working together can paint the apartment in 4 hours.
Step-by-step explanation:
Given:
Time taken by new crew of painters to paint an apartment = 12 hours
Time taken by experience crew of painters to paint an apartment = 6 hours
To find the time taken by them to paint the apartment working together.
Solution:
Using unitary method to determine their 1 hour work.
In 12 hours the new crew can paint = 1 apartment
So,in 1 hours the new crew can paint = [tex]\frac{1}{12}[/tex] of the apartment
In 6 hours the experience crew can paint = 1 apartment
So,in 1 hour the experience crew can paint = [tex]\frac{1}{6}[/tex] of the apartment
Now, the new crew and the experience crew are working together.
So, in 1 hour, they can paint :
⇒ [tex]\frac{1}{12}+\frac{1}{6}[/tex]
Taking LCD = 12, we will add fraction.
⇒ [tex]\frac{1}{12}+\frac{2}{12}[/tex]
⇒ [tex]\frac{3}{12}[/tex]
Simplifying fraction we have:
⇒ [tex]\frac{1}{4}[/tex] of an apartment
Again using unitary method to determine the time taken by them working together to paint the whole apartment.
They can paint [tex]\frac{1}{4}[/tex] of an apartment in 1 hour.
To paint 1 apartment they will take = [tex]\frac{1}{\frac{1}{4}}=4[/tex] hours
Thus, the new crew and the experience crew working together can paint the apartment in 4 hours.
A store receives a shipment of pens and pencils. Each box contains 75 items There are 23 boxes of pencils and 18 boxes of pens. How many pens and pencils did the store receive?
The store received 1725 pencils and 1350 pens as of the given conditions.
Given that,
A store receives a shipment of pens and pencils. Each box contains 75 items There are 23 boxes of pencils and 18 boxes of pens. How many pens and pencils the store receive is to be determined.
What is arithmetic?
In mathematics, it deals with numbers of operations according to the statements. There are four major arithmetic operators, addition, subtraction, multiplication and division,
Here,
Each box contains 75 items,
The number of item in 23 pencil boxes = 75 × 23 = 1725 items,
The number of items in 18 pencil boxes = 18 × 75 = 1350 items.
Thus, the store received 1725 pencils and 1350 pens as of the given conditions.
Learn more about arithmetic here:
https://brainly.com/question/11424589
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Find the equation of a line that has a slope of –4,
and includes the point (4, –9).
A. y = –4x – 32
B. y = –4x – 25
C. y = –4x – 8
D. y = –4x + 7
E. y = –4x + 40
Answer:
D
Step-by-step explanation:
Finding y int by substituting the points given (say y int is x)
(-9) = 4(-4) + x
(-9) = -16 + x
x = 7
With the y int, you can now write the equation
y = -4x + 7
The equation of the line will be y = –4x + 7 having a slope of –4 that passes through the point (4, –9). Option D is correct.
To find the equation of a line that has a slope of –4, and includes the point (4, –9), we can use the point-slope form of a linear equation: y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.
Plugging in our values, we get y - (-9) = -4(x - 4). Simplifying this, we have:
y + 9 = -4x + 16Subtract 9 from both sides to solve for y:
y = -4x + 16 - 9y = -4x + 7Thus, the correct equation is y = –4x + 7.
Hence, D. is the correct option.
A researcher has two percentages and wants to know if the percentages are statistically different. The researcher calculates the z value and finds that it is 4.21. This means that the two percentages: A) Are the same. B) Are not statistically different. C) Have a 421 percent chance of not being different. D) Are statistically different.
Answer:
[tex]p_v =2*P(Z>4.21) =2.55x10^{-5}[/tex]
And we can use the following excel code to find it:"=2*(1-NORM.DIST(4.21,0,1,TRUE)) "
With the p value obtained and using the significance level assumed for example[tex]\alpha=0.05[/tex] we have [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the percentage 1 is significantly different from the percentage 2.
D) Are statistically different.
Step-by-step explanation:
The system of hypothesis on this case are:
Null hypothesis: [tex]\mu_1 = \mu_2[/tex]
Alternative hypothesis: [tex]\mu_1 \neq \mu_2[/tex]
Or equivalently:
Null hypothesis: [tex]\mu_1 - \mu_2 = 0[/tex]
Alternative hypothesis: [tex]\mu_1 -\mu_2\neq 0[/tex]
Where [tex]\mu_1[/tex] and [tex]\mu_2[/tex] represent the percentages that we want to test on this case.
The statistic calculated is on this case was Z=4.21. Since we are conducting a two tailed test the p value can be founded on this way.
[tex]p_v =2*P(Z>4.21) =2.55x10^{-5}[/tex]
And we can use the following excel code to find it:"=2*(1-NORM.DIST(4.21,0,1,TRUE)) "
With the p value obtained and using the significance level assumed for example[tex]\alpha=0.05[/tex] we have [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the percentage 1 is significantly different from the percentage 2.
And the best option on this case would be:
D) Are statistically different.
The gas mileage for a certain model of car is known to have a standard deviation of 6 mi/gallon. A simple random sample of 64 cars of this model is chosen and found to have a mean gas mileage of 27.5 mi/gallon. Construct a 97.5% confidence interval for the mean gas mileage for this car model.
Answer: confidence interval = 27.5 +/- 1.68
= ( 25.82, 29.18)
Step-by-step explanation:
Given;
Number of samples n = 64
Standard deviation r = 6mi/gallon
Mean x = 27.5mi/gallon
Confidence interval of 97.5%
Z' = t(0.0125) = 2.24
Confidence interval = x +/- Z'(r/√n)
= 27.5 +/- 2.24(6/√64)
= 27.5 +/- 1.68
= ( 25.82, 29.18)