Answer: it would take 3.1 months
Step-by-step explanation:
Let m represent the number of months that that it will take either of the runners to have the same amount of money in their accounts.
Let y represent the total amount that runner 1 saves for m months
Let z represent the total amount that runner 2 saves for m months
The first runner has $112 in savings, received a $45 gift from a friend, and will save $25 each month. This means that the total amount that he will save for m months would be
y = 112 + 45 + 25m
y = 157 + 25m
The second runner has $50 in savings and will save $60 each month. This means that the total amount that he will save for m months would be
z = 50 + 60m
To determine the number of months before the amount in the account of both runners becomes the same, we would equate y to z. It becomes
157 + 25m = 50 + 60m
157 - 50 = 60m - 25m
35m = 107
m = 107/35
m = 3.06
Does there exist a di↵erentiable function g : [0, 1] R such that g'(x) = f(x) for all x 2 [0, 1]? Justify your answer
Answer:
No; Because g'(0) ≠ g'(1), i.e. 0≠2, then this function is not differentiable for g:[0,1]→R
Step-by-step explanation:
Assuming: the function is [tex]f(x)=x^{2}[/tex] in [0,1]
And rewriting it for the sake of clarity:
Does there exist a differentiable function g : [0, 1] →R such that g'(x) = f(x) for all g(x)=x² ∈ [0, 1]? Justify your answer
1) A function is considered to be differentiable if, and only if both derivatives (right and left ones) do exist and have the same value. In this case, for the Domain [0,1]:
[tex]g'(0)=g'(1)[/tex]
2) Examining it, the Domain for this set is smaller than the Real Set, since it is [0,1]
The limit to the left
[tex]g(x)=x^{2}\\g'(x)=2x\\ g'(0)=2(0) \Rightarrow g'(0)=0[/tex]
[tex]g(x)=x^{2}\\g'(x)=2x\\ g'(1)=2(1) \Rightarrow g'(1)=2[/tex]
g'(x)=f(x) then g'(0)=f(0) and g'(1)=f(1)
3) Since g'(0) ≠ g'(1), i.e. 0≠2, then this function is not differentiable for g:[0,1]→R
Because this is the same as to calculate the limit from the left and right side, of g(x).
[tex]f'(c)=\lim_{x\rightarrow c}\left [\frac{f(b)-f(a)}{b-a} \right ]\\\\g'(0)=\lim_{x\rightarrow 0}\left [\frac{g(b)-g(a)}{b-a} \right ]\\\\g'(1)=\lim_{x\rightarrow 1}\left [\frac{g(b)-g(a)}{b-a} \right ][/tex]
This is what the Bilateral Theorem says:
[tex]\lim_{x\rightarrow c^{-}}f(x)=L\Leftrightarrow \lim_{x\rightarrow c^{+}}f(x)=L\:and\:\lim_{x\rightarrow c^{-}}f(x)=L[/tex]
1) Find the minimum and maximum values for the function with the given domain interval.
minimum value = 7; maximum value = 8
minimum value = 0; maximum value = 7
minimum value = 0; maximum value = none
minimum value = none; maximum value = 8
minimum value = 0; maximum value = 8
Answer:
"minimum value = 0; maximum value = 8"
Step-by-step explanation:
This is the absolute value function, which returns a positive value for any numbers (positive or negative).
For example,
| -9 | = 9
| 9 | = 9
| 0 | = 0
Now, the domain is from -8 to 7 and we want to find max and min value that we can get from this function.
If we look closely, putting 7 into x won't give us max value as putting -8 would do, because:
|7| = 7
|-8| = 8
So, putting -8 would give us max value of 8 for the function.
Now, we can't get any min values that are negative, because the function doesn't return any negative values. So the lowest value would definitely be 0!
|0| = 0
and
ex: |-2| = 2 (bigger), |-5| = 5 (even bigger).
So,
Min Value = 0
Max Value = 8
Answer:
minimum value = 0; maximum value = 8
Step-by-step explanation:
The function [tex]f(x)[/tex] is an absolute value function, which means that for negative values in it's domain it gives positive values of [tex]f(x)[/tex], and therefore it's minimum value is 0.
In the given domain interval the maximum value of the function is 8 because [tex]f(-8)=8[/tex].
If the square root of the length of the hypotenuse of a right triangle is 2 units, what is the sum of the squares of the length of the two other sides?
Answer:16
Step-by-step explanation:
Fatima works at a bakery. She places 5 candied flowers on top of each cupcake she decorates. She Will decorate 2 dozen cupcakes today and 2 dozen tomorrow how many candied flower will Fatima use in these two days
Answer: 240 candied flowers
Step-by-step explanation:
She places 5 candied flowers on top of each cupcake she decorates. She will decorate 2 dozen cupcakes today. A dozen cupcakes is 12. 2 dozen cupcakes would be 24. Total number of candied flowers that she will place on top of each cupcake today would be 24 × 5 = 120 candied flowers.
She will also decorate 2 dozens tomorrow. Total number of candied flowers that she will place on top of each cupcake tomorrow would be 24 × 5 = 120 candied flowers
Total number if candied flowers that Fatima will use in 2 days would be 120 + 120 = 240
Suppose that the distribution is bell-shaped. If approximately 99.7% of the lifetimes lie between 568 hours and 1066 hours, then the approximate value of the standard deviation for the distribution, according to the empirical rule, is .
Answer:
[tex]\sigma =\frac{478}{6}=79.667[/tex]
Step-by-step explanation:
The empirical rule, also referred to as "the three-sigma rule or 68-95-99.7 rule, is a statistical rule which states that for a normal distribution, almost all data falls within three standard deviations (denoted by σ) of the mean (denoted by µ)". The empirical rule shows that 68% falls within the first standard deviation (µ ± σ), 95% within the first two standard deviations (µ ± 2σ), and 99.7% within the first three standard deviations (µ ± 3σ).
And on this case since we are within 3 deviations (because we have 99.7% of the data between 568 and 1066hours), the result obtained using the z score agrees with the empirical rule.
So on this case we can find the standard deviation on this ways:
[tex]\mu -3\sigma = 568[/tex] (1)
[tex]\mu +3\sigma = 1066[/tex] (2)
If we subtract conditions (2) and (1) we got:
[tex]1066-588 =\mu +3\sigma -\mu +3\sigma[/tex]
[tex]478= 6\sigma[/tex]
[tex]\sigma =\frac{478}{6}=79.667[/tex]
John rides his bike to work each day. The distance between his house and his work is approximately 6.5 miles, and it takes him on average 45 minutes to get there on his bike. In order to compare the approximate speed of his bike to that of a car, he determines his average speed on the bike in miles per hour.
Which of the following values most likely represents the value John determined to be his speed on the bike in miles per hour?
A
9.0 mph
B
8.7 mph
C
8.66 mph
D
8.667 mph
Answer:
D) 8.667 mph
Step-by-step explanation:
Given: Distance= 6.5 miles
Times= 45 minutes
First, convert the time into hours as we need to find speed in the unit of mph.
We know, 1 hour= 60 minutes
∴ Time= [tex]\frac{45}{60} = 0.75\ h[/tex]
Now, find the speed of John´s bike
Speed= [tex]\frac{distance}{time}[/tex]
⇒ Speed= [tex]\frac{6.5}{0.75} = 8.667\ mph[/tex]
∴ Speed of John´s bike is 8.667 mph
A fire department's longest ladder is 110 feet, and the maximum height it can be used for is 100 feet. What is the angle that ladder makes with the ground at the maximum height?
Check the picture below.
Answer: The angle that the ladder makes with the ground at the maximum height is 65.37 degrees
Step-by-step explanation:
The triangle ABC is formed by the ladder and the wall is shown in the attached photo.
The angle that the ladder makes with the ground at the maximum height is represented as #. To determine #, we will apply trigonometric ratio
Sin # = 0pposite side / hypotenuse.
Hypotenuse = 110
Opposite side = 100
Sin# = 100/110 = 0.909
# = Sin^(-1)0.909
# = 65.37
Please help!
Which is a solution to the linear function 5x – 2y = 12?
A. (3, –1)
B. (2, 1)
C. (3, 1)
D. (2, –1)
For every positive integer n, the nth term of sequence is given by an= 1/n - 1/(n+1). What is the sum of the first 100 terms?
(a) 1
(b) 0
(c) 25
(d) 99/100
(e) 100/101
Step-by-step explanation:
We need to find um of the first 100 terms of
[tex]\frac{1}{n}-\frac{1}{n+1}[/tex]
That is
[tex]\texttt{Sum = }\frac{1}{1}-\frac{1}{1+1}+\frac{1}{2}-\frac{1}{2+1}+\frac{1}{3}-\frac{1}{3+1}.....+\frac{1}{100}-\frac{1}{100+1}\\\\\texttt{Sum = }\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}.....+\frac{1}{100}-\frac{1}{101}\\\\\texttt{Sum = }\frac{1}{1}-\frac{1}{101}\\\\\texttt{Sum = }\frac{101-1}{101\times 1}\\\\\texttt{Sum = }\frac{100}{101}[/tex]
Option E is the correct answer.
Final answer:
The sum of the first 100 terms of the sequence an = 1/n - 1/(n+1) is 100/101 because it's a telescoping series where almost all terms cancel each other out except the very first and the very last term.
Explanation:
The student's question involves finding the sum of the first 100 terms of the sequence an = 1/n - 1/(n+1). To find this sum, we can notice that many terms will cancel each other out when we add up the sequence. This is because the sequence is telescoping. Let's illustrate this with the first few terms:
a1 = 1 - 1/2
a2 = 1/2 - 1/3
a3 = 1/3 - 1/4
...
a99 = 1/99 - 1/100
a100 = 1/100 - 1/101
When we add all these up, notice that every negative term cancels out with the positive term that precedes it, except for the very first term, which is 1, and the very last negative term, which is -1/101. Hence, the sum is 1 - 1/101 which simplifies to 100/101. Therefore, the correct answer is (e) 100/101.
Student scores on exams given by a certain instructor have mean 74 and standard deviation 14. This instructor is about to give two exams, one to a class of size 25 and the other to a class of size 64. Approximate the probability that the average test score in the class of size 25 exceeds 80.
Answer:
[tex]P(\bar X >80)=P(Z>2.143)=1-P(z<2.143)=1-0.984=0.016[/tex]
Step-by-step explanation:
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
Let X the random variable that represent the Student scores on exams given by a certain instructor, we know that X have the following distribution:
[tex]X \sim N(\mu=74, \sigma=14)[/tex]
The sampling distribution for the sample mean is given by:
[tex]\bar X \sim N(\mu,\frac{\sigma}{\sqrt{n}})[/tex]
The deduction is explained below we have this:
[tex]E(\bar X)= E(\sum_{i=1}^{n}\frac{x_i}{n})= \sum_{i=1}^n \frac{E(x_i)}{n}= \frac{n\mu}{n}=\mu[/tex]
[tex]Var(\bar X)=Var(\sum_{i=1}^{n}\frac{x_i}{n})= \frac{1}{n^2}\sum_{i=1}^n Var(x_i)[/tex]
Since the variance for each individual observation is [tex]Var(x_i)=\sigma^2 [/tex] then:
[tex]Var(\bar X)=\frac{n \sigma^2}{n^2}=\frac{\sigma}{n}[/tex]
And then for this special case:
[tex]\bar X \sim N(74,\frac{14}{\sqrt{25}}=2.8)[/tex]
We are interested on this probability:
[tex]P(\bar X >80)[/tex]
And we have already found the probability distribution for the sample mean on part a. So on this case we can use the z score formula given by:
[tex]z=\frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
Applying this we have the following result:
[tex]P(\bar X >80)=P(Z>\frac{80-74}{\frac{14}{\sqrt{25}}})=P(Z>2.143)[/tex]
And using the normal standard distribution, Excel or a calculator we find this:
[tex]P(Z>2.143)=1-P(z<2.143)=1-0.984=0.016[/tex]
Final answer:
Using the Central Limit Theorem and the z-score formula, we calculate that the approximate probability that the average test score in the class of size 25 exceeds 80 is approximately 1.62%.
Explanation:
To approximate the probability that the average test score in the class of size 25 exceeds 80, we can use the Central Limit Theorem which tells us that the sampling distribution of the sample mean will be approximately normally distributed if the sample size is large enough (typically n ≥ 30 is considered sufficient, but we can still use this for a sample of 25 when the population distribution is not overly skewed).
The formula for the z-score of a sample mean is:
z = (X - μ) / (σ / √n)
where X is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. Given the population mean μ = 74, population standard deviation σ = 14, and sample size n = 25, we can calculate the z-score for a sample mean of 80.
Using these values:
z = (80 - 74) / (14 / √25) = (6) / (14 / 5) = 6 / 2.8 = 2.14
Now, we need to find the probability corresponding to a z-score of 2.14. We check the standard normal distribution table or use a calculator with normal distribution functions to find that the area to the left of z = 2.14 is approximately 0.9838. The probability that the average is above 80 is the area to the right of 2.14, so we subtract this value from 1.
Probability = 1 - 0.9838 = 0.0162
The approximate probability that the average test score in the class of size 25 exceeds 80 is approximately 0.0162, or 1.62%.
How do you do this question?
Answer:
D) dy/dx > 0 and d²y/dx² > 0
Step-by-step explanation:
Use implicit differentiation to find dy/dx and d²y/dx².
x²y³ = 576
x² (3y² dy/dx) + (2x) y³ = 0
3x²y² dy/dx = -2xy³
3x dy/dx = -2y
dy/dx = -2y / (3x)
d²y/dx² = [ (3x) (-2 dy/dx) − (-2y) (3) ] / (3x)²
d²y/dx² = (-6x dy/dx + 6y) / (9x²)
d²y/dx² = (-6x (-2y / (3x)) + 6y) / (9x²)
d²y/dx² = (4y + 6y) / (9x²)
d²y/dx² = 10y / (9x²)
Evaluating each at (-3, 4):
dy/dx = -2(4) / (3(-3))
dy/dx = 8/9
d²y/dx² = 10(4) / (9(-3)²)
d²y/dx² = 40/81
Both are positive.
Your school is sponsoring a pancake dinner to raise money for a field trip. You estimate that 200 adults and 250 children will attend. Let x represent the cost of an adult ticket and y represent the cost of a child ticket.
Write an equation that can be used to find what ticket prices to set in order to raise $3800
Show your work
Answer:
Step-by-step explanation:
Let x represent the cost of an adult ticket and
Let y represent the cost of a child ticket.
Your school is sponsoring a pancake dinner to raise money for a field trip. You estimate that 200 adults and 250 children will attend.
The equation that can be used to find what ticket prices to set in order to raise $3800 would be
200x + 250y = 3800
The equation that can be used to find what ticket prices to set in order to raise $3800 is [tex]3800=200x+250y[/tex].
What is an equation?An equation is formed when two equal expressions are equated together with the help of an equal sign '='.
As it is given that the cost of an adult ticket is x while the number of adult tickets sold was 200. Similarly, the cost of a child ticket is y while the number of child tickets sold will be 250. And the total money that is needed to be raised is $3800, therefore, the equation can be written as,
Total amount= Total amount of Adult Tickets + Total amount of Child Ticket
[tex]\$3,800 = (\$x \times 200)+(\$y \times 250)\\\\3800=200x+250y[/tex]
Hence, the equation that can be used to find what ticket prices to set in order to raise $3800 is [tex]3800=200x+250y[/tex].
Learn more about Equation:
https://brainly.com/question/2263981
Lloyd's Cereal company packages cereal in 1 pound boxes (16 ounces). A sample of 16 boxes is selected at random from the production line every hour, and if the average weight is less than 15 ounces, the machine is adjusted to increase the amount of cereal dispensed. If the mean for 1 hour is 1 pound and the standard deviation is 0.1 pound, what is the probability that the amount dispensed per box will have to be increased?
Answer:
The probability that the amount dispensed per box will have to be increased is 0.0062.
Step-by-step explanation:
Consider the provided information.
Sample of 16 boxes is selected at random.
If the mean for 1 hour is 1 pound and the standard deviation is 0.1
1 Pound = 16 ounces , then 0.1 Pound = 16/10 = 1.6 ounces
Thus: μ = 16 ounces and σ = 1.6 ounces.
Compute the test statistic [tex]z=\frac{\bar x-\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
[tex]z=\frac{15-16}{\frac{1.6}{\sqrt{16}}}[/tex]
[tex]z=\frac{-1}{\frac{1.6}{4}}[/tex]
[tex]z=\frac{-1}{0.4}[/tex]
[tex]z=-2.5[/tex]
By using the table.
P value = P(Z<-250) = 0.0062
Thus, the probability that the amount dispensed per box will have to be increased is 0.0062.
Breandan makes a cranberry orange drink by mixing 15 cups of orange juice. If he uses 27 cups of orange juice how many cups of orange juice how many cranberry juice should he use to make.
To keep the taste consistent, Brendan should use 18 cups of cranberry juice to mix with 27 cups of orange juice, preserving the original 3:2 juice ratio.
Brendan's original mixture was 15 cups of orange juice to 10 cups of cranberry juice. This creates a ratio of 15:10, which simplifies to 3:2 when divided by 5. To maintain the same taste, Brendan will want to keep the same ratio.
Now to calculate the amount of cranberry juice needed for 27 cups of orange juice, we set up a proportion
Set up a proportion to find the unknown value (x), representing the amount of cranberry juice:
3/2 = 27/x
Cross-multiply to solve for x:
3x = 2 x 27
3x = 54
Divide both sides by 3 to solve for x:
x = 54 / 3
x = 18
Brendan should use 18 cups of cranberry juice to mix with 27 cups of orange juice to keep the taste of the drink consistent.
The complete question is:
Brendan makes a cranberry-orange drink by mixing 15 cups of orange juice with 10 cups of cranberry juice. If he uses 27 cups of orange juice, how many cups of cranberry juice should he use in order for the drink to taste the same?
The radius r(t)r(t)r, (, t, )of a sphere is increasing at a rate of 7.57.57, point, 5 meters per minute. At a certain instant t_0t 0 t, start subscript, 0, end subscript, the radius is 555 meters. What is the rate of change of the surface area S(t)S(t)S, (, t, )of the sphere at that instant?
Answer:
300pi
Step-by-step explanation:
Final answer:
The rate of change of the surface area of the sphere at that instant is 942.48 meters squared per minute.
Explanation:
To find the rate of change of the surface area S(t)S(t)S, (, t, )of the sphere at that instant, we need to differentiate the surface area formula with respect to time and then substitute the given values.
The formula for the surface area of a sphere is [tex]S = 4\pi r^2.[/tex]
Taking the derivative with respect to time, we have dS/dt = 8πr(dr/dt).
Given that dr/dt = 7.5 meters per minute and r = 5 meters, we can substitute these values into the derivative formula to find the rate of change of the surface area at that instant.
= dS/dt = 8π(5)(7.5)
= 300π
= 942.48 meters squared per minute.
Plssssssssssssssssssss Answer this is Major?
This activity will help you meet these educational goals:
You will create a quadratic function to model the area of a bean-bag toss carnival game, and then graph it and examine its key features.
Your woodworking class is going to make games for the school carnival. You are in charge of making a rectangular game board for a bean bag toss. The length and width of the board have a specific relationship that is shown by the algebraic expressions in the image, which represents a possible finished game board. The units are in inches.
Part A
Enter the correct answer in the box.
Use the expressions that represent the length and width of the game board to write an equation that models the area of the figure. Let y represent the area, and write your answer in the form y = ax2 + bx + c, where a, b, and c are real numbers.
Part B
Graph the equation you wrote in part A. Adjust the zoom of the graphing window so the vertex, x-intercepts, and y-intercept can be seen.
Part C
The graph of a quadratic equation always has an extreme location (maximum or minimum). State whether the parabola opens upward or downward, whether it has a maximum or a minimum, and what the coordinates of that point are. Use the pointer tool to approximate the coordinates of this extreme location to the nearest whole number.
Part D
According to the graph, what is the maximum possible area of the game board? Give your answer to the nearest whole number. (Assume that the maximum area is not reduced by the open hole in the game board.)
Part E
Type the correct answer in each box.
Use the original expressions for the length and width, and substitute the x-coordinate from the extreme location. What are the length and width of the game board at the extreme location?
The length is ________________inches, and the width is ____________
inches.
Part F
What type of quadrilateral will be formed when the game board covers the maximum possible area?
Part G
Suppose the carnival director asks you to create a game board that is 1,120 square inches. Find the dimensions that would meet this request by setting the area equation equal to 1,120, solving for x, and substituting x into the expressions for the length and width. As before, assume the open hole in the game board does not affect the area calculation.
Part H
When you solved the area equation for x, did any extraneous solutions result? Describe how an extraneous solution would arise in this situation.
Part I
What method of solving quadratics did you use to solve the equation set equal to 1,120? Why did you choose this method? Discuss the usefulness of other methods of solving quadratics as they pertain to this scenario. Use this resource to help refresh your memory on methods for solving quadratic equations.
Answer:
See below because there are 9 parts (A through I)
Explanation:
Part A: write an equation that models the area of the figure. Let y represent the area, and write your answer in the form y = ax2 + bx + c.
The figure shows a rectangular table with these dimensions:
Length: - x + 64Witdth: x + 4The area of a rectangle is width × length:
[tex](x + 4)\times (-x+64)[/tex]Use distributive property:
[tex]x\cdot (-x)+x\cdot(64)+4\cdot (-x)+4\cdot (64)=-x^2+64x-4x+256[/tex]Simplify:
[tex]-x^2+64x-4x+256=-x^2+60x+256[/tex]Part B. Graph the equation you wrote in part A. Adjust the zoom of the graphing window so the vertex, x-intercepts, and y-intercept can be seen.
1. Factor the equation:
Common factor - 1:[tex]-x^2+60x+256=-(x^2-60x-256)[/tex]
Find two numbers that add - 60 and whose product is -256. Theyb are -64 and + 4[tex]-(x-64)(x+4)[/tex]
2. Find the roots:
Equal the expression to zero:
[tex]-(x-64)(x+4)=0\\ \\ x-64=0\implies x=64\\ \\ x+4=0\implies x=-4[/tex]
Those are the x-intercepts: (-4,0) and (64,0)
3. Find the symmetry axis:
The simmetry axis is the line x = the middle value between the two roots:
[tex]x=(64-4)/2=60/2=30[/tex]
4. Find the vertex
The vertex has x-coordinate equal to the x axis (30 in this case).
Substitute in the equation of find the y-coordinate:
[tex]y=-(30-64)(30+4)=-(-34)(34)=1,156[/tex]
Hence, the vertex is (30, 1,156)
5. Find the y-intercept
Make x = 0
[tex]y=-(x^2-60x-256)=-(0-256)=256[/tex]
Hence, the y-intercept is (0, 256)
With the x-incercepts, the y-intercept, the axis of symmetry, and the vertex, you can sketch the graph.
You can see now the graph in the attached figure
Part C. Extreme location of the graph
The graph shows that the parabola opens downward. That is due to the fact that the coefficient of the leading term (x²) is negative.
The parabola starts in the second quadrant. starts growing, crosses the x-axis at (-4,0), crosses the y-axis at (0,256), reaches the maximum value at (30, 1156), and then decreases toward the fouth quadrant, crossing the x-axis at (64,0).
Thus the vertex is a maximun, and the coordinates of the maximum are (30, 1156).
Part D. According to the graph, what is the maximum possible area of the game board? Give your answer to the nearest whole number. (Assume that the maximum area is not reduced by the open hole in the game board.)
The maximum possible area of the game is the maximum value of the function y = -x² + 60x + 256.
This value was calculated as y = 1156.
Part E. Use the original expressions for the length and width, and substitute the x-coordinate from the extreme location. What are the length and width of the game board at the extreme location?
The length is:
length = - x + 64 inchesx = 30length = - 30 + 64 = 34 inchesThe width is:
width = x + 4x = 30width = 30 + 4 = 34 inchesPart F. What type of quadrilateral will be formed when the game board covers the maximum possible area?
Since the length and the width are equal, the quadrilateral is a square.
Part G. Suppose the carnival director asks you to create a game board that is 1,120 square inches. Find the dimensions that would meet this request by setting the area equation equal to 1,120, solving for x, and substituting x into the expressions for the length and width.
[tex]y=-x^2+60x+256\\ \\ 1,120=-x^2+60x+256\\ \\ x^2-60x-256+1120=0\\ \\ x^2-60x+864=0[/tex]
Factor:
Find two numbers whose sum is - 60 and the product os 864. They are -24 and - 34:
[tex]x^2-60x+864=(x-24)(x-36)[/tex]
Use the zero product rule:
[tex](x-24)(x-36)=0\\ \\ x-24=0\implies x=24\\ \\ x-36=0\implies x=36[/tex]
Now substitute to find the dimensions:
x = 36
length = - x + 64length = - 36 + 64 = 28width = x + 4 = 36 + 4 = 40Hence, legth = 28, width = 40
x = 24
length = - x + 64 = -24 + 64 = 40width = x + 4 = 24 + 4 = 28Part H. When you solved the area equation for x, did any extraneous solutions result? Describe how an extraneous solution would arise in this situation.
The two solutions are valid (non extraneous) because both leads to positive real dimensions for which the areas can be 1,120 in².
28×40 = 1,12040×28 = 1,120An extraneous solution could arise if you try to find areas for which x is greater than or equal to 64, because in that case - x + 64 would be zero or negative and dimensions must be positive.
For the same reason, also an extraneous solution would arise if you try to fix areas for which x is less than or equal to - 4.
So, the domain of your function has to be - 4 < x < 64.
Part I. What method of solving quadratics did you use to solve the equation set equal to 1,120? Why did you choose this method?
The method use was factoring.
Discuss the usefulness of other methods of solving quadratics as they pertain to this scenario.
The other importants methods are graphical and the quadratic equation.
For graphical method you graph your parabola and find the values of x that sitisfies the area searched (value of y).
The quadratic equation gives the y-values (areas) without factoring:
[tex]\frac{-b+/-\sqrt{b^2-4(a)(c)} }{2(a)}[/tex]
Roberto shares a bag of almonds with 2 friends. He shares 1/8 bag with Jeremy and 2/8 bag with Emily. He eats 3/8 bag of the almonds himself. What fraction of the almonds do Roberto and his friends eat?
Answer:The fraction of the almonds that Roberto and his friends ate is 3/4
Step-by-step explanation:
Let x represent the total number of almonds in the bag initially. He shares 1/8 bag with Jeremy. This means that the amount of almonds that he gave to Jeremy is 1/8 × x = x/8
He shares 2/8 bag with Emily. This means that the amount of almonds that he gave to Emily is 2/8 × x = 2x/8
He eats 3/8 bag of the almonds himself. This means that the amount of almonds that he ate is 3/8 × x = 3x/8
Total number of almonds that Robert and his friends ate would be
x/8 + 2x/8 + 3x/8 = 6x/8 = 3x/4
The fraction of the almonds that Roberto and his friends ate would be
(3x/4)/x = 3/4
Roberto and his friends eat a total of 3/4 of a bag of almonds, calculated by adding the fractions of the bag each person consumed.
Explanation:The student is asking how to calculate the total fraction of a bag of almonds eaten by Roberto and his friends. Roberto shares 1/8 of the bag with Jeremy, 2/8 of the bag with Emily, and eats 3/8 of the bag himself. To find the total fraction consumed, we add these fractions together:
1/8 (Jeremy) + 2/8 (Emily) + 3/8 (Roberto) = 6/8
Since 2/8 can be simplified to 1/4, and 6/8 can be simplified to 3/4, the total fraction of the almonds eaten by Roberto and his friends is 3/4 of the bag.
The length, width, and height of a rectangular prism is a, (a-13), and (a+13) respectively, which statement best describes (a-13)a. the width of the rectangular prism is 13 units less than the lengthb. the height of the rectangular prism is 13 units more than the lenghtc. the width of the rectangular prism is 13 units less than the heightd. the length of the rectangular prism is 13 units less than the weight
Answer:
a. The width of the rectangular prism is 13 units less than the length.
Step-by-step explanation:
From the information given we know that
a = length of rectangular prism and
[tex]a - 13[/tex] = width of rectangular prism
An expression like [tex]a - 13[/tex] means that we need to subtract 13 units from the length to obtain the width of the rectangular prism.
Therefore,
The width of the rectangular prism is 13 units less than the length.
A clothing store is selling a shirt for a discounted price of $43.61. If the discount is 11%, what was the original price, in dollars, of the shirt? Do not include units in your answer.
A clothing store is selling a shirt for a discounted price of $43.61 . The original price of the shirt was approximately $49.01.
Explanation:To find the original price of the shirt, we can use the formula: Original Price = Discounted Price / (1 - Discount Rate). In this case, the discounted price is $43.61 and the discount rate is 11%, or 0.11. Plugging these values into the formula, we get: Original Price = 43.61 / (1 - 0.11) = 43.61 / 0.89 ≈ 49.01. Therefore, the original price of the shirt was approximately $49.01.
Learn more about original price of a discounted item here:https://brainly.com/question/32856381
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Which coordinate divides the directed line segment from −10 at J to 23 at K in the ratio of 2 to 1?
1
11
12
Answer:
12
Step-by-step explanation:
The difference of the two coordinates is ...
23 -(-10) = 33
The desired coordinate is 2/3 of that length from J, so is ...
J + (2/3)·33 = J +22 = -10 +22 = 12
The desired coordinate is 12.
Please help I'm stuck :(
You are going on a trip to see friends in Georgia. At 8 am, you have driven 50 miles. By noon, you have driven 300 miles. At the same rate, how far have you driven by 2 pm?
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speed = distance/ time
speed = 250/4
speed = 62.5 mph
distance = speed x time
distance = 62.5 x 2
distance = 125 miles
Your answer is 125 miles
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Larry studied 2 1/4 hours Monday. He studied 2 5/6 hours Tuesday. Write an addition sentence to show how many hours he spent studying Monday and Tuesday.
Final answer:
Larry studied a total of 5 1/12 hours on Monday and Tuesday. To find this, convert the mixed numbers to improper fractions, find a common denominator, add the fractions together, and simplify to get the final sum.a
Explanation:
To calculate the total amount of time Larry spent studying on Monday and Tuesday, we need to add the hours together:
1. Monday: 2 1/4 hours
2. Tuesday: 2 5/6 hours
Let's convert these mixed numbers to improper fractions to simplify the addition:
Convert 2 1/4 to an improper fraction: 2 1/4 = (2×4)+1/4 = 9/4.Convert 2 5/6 to an improper fraction: 2 5/6 = (2×6)+5/6 = 17/6.Next, we find a common denominator, which is 12, and rewrite the fractions:
Rewrite 9/4 as a fraction with a denominator of 12: 9/4 = (9×3)/(4×3) = 27/12.Rewrite 17/6 as a fraction with a denominator of 12: 17/6 = (17×2)/(6×2) = 34/12.Now that they have the same denominator, we can add them together:
27/12 + 34/12 = 61/12To simplify, divide 61 by 12, which is 5 with a remainder of 1. Thus, the mixed number is 5 1/12. Therefore, the addition sentence to show how many hours Larry spent studying Monday and Tuesday is:
2 1/4 hours + 2 5/6 hours = 5 1/12 hours.An Internet service provider allows a certain number of free hours each month and then charges for each additional hour used. Wells, Ted, and Vino each have separate accounts. This month the total hours used by Wells and Ted was 105, and each used all of their free hours. Their total cost was $\$10$. Vino used 105 hours by himself and had to pay $\$26$. What is the number of cents charged for each extra hour
To find the cost charged for each extra hour, we can set up a system of equations using the given information. By solving these equations, we can determine the value of the number of free hours provided (A) and the cost charged for each extra hour (z).
Explanation:To find the cost charged for each extra hour, we need to set up a system of equations using the given information. Let:
A = number of free hours providedx = hours used by Wells and Ted (105 hours in total)y = number of extra hours used by Wells and Tedz = cost charged for each extra hourw = hours used by Vino (105 hours)From the given information, we can set up the following equations:
2A + zy = 10 (equation 1)
w + z(w-1) = 26 (equation 2)
Since Wells and Ted used all of their free hours (A), the number of extra hours used by them would be x-A (105-A). So, we can substitute y = 105-A in equation 1:
2A + z(105-A) = 10
Simplifying this equation, we get:
2A + 105z - zA = 10
Combining like terms:
A(2-z) + 105z = 10
Solving equation 2 for z:
z = (26-w)/(w-1)
Substituting this value of z in equation 3:
A(2 - (26-w)/(w-1)) + 105(26-w)/(w-1) = 10
Simplifying this equation will give us the value of A, and from there, we can find the value of z.
Two number cubes with faces numbered 1 to 6 are rolled. What is the probability that that sum of the rolls is 8.
A. 1/12
B. 1/11
C. 1/9
D. 5/36
E. 1/6
Answer:
D
Step-by-step explanation:
Firstly, we need to know the number of total possible results. We can do this by placing the first die in horizontal band and the second die in vertical band.
The total number of results would be 36 results.
Now, to get the number of 8s
The possible sums that can give 8 is 2 and 6, 2and 5 and 3 with 4 and 4.
All are possible two times asides the 4 and 4 that could only show one time.
This means as we can have 2 and 6 we can also have 6 and 2
The total number of expected results is thus: 5/36
Final answer:
The probability of rolling a sum of 8 with two number cubes is 5/36.
Explanation:
To find the probability of rolling a sum of 8 with two number cubes, we need to look at all the possible combinations that result in an 8.
The number cubes (dice) each have faces numbered from 1 to 6.
We can roll a (2,6), (6,2), (3,5), (5,3), (4,4) to get a sum of 8.
This gives us a total of 5 favorable outcomes.
Since each die has 6 faces, the number of possible outcomes when rolling two dice is 6 x 6, which equals 36.
To get the probability of the sum being 8, we divide the number of favorable outcomes (5) by the total number of possible outcomes (36).
This calculation gives us the probability 5/36.
"Bill received $12 to feed a neighbor's cat for 3 days. At this pay rate, how many
days will he have to feed the cat to earn $40? The neighbor's family is going on
vacation for 3 weeks next summer. Bill wants to earn enough money to buy a CD
player that costs $89. Will he have enough money? Explain.
Answer:
Step-by-step explanation:
Bill received $12 to feed a neighbor's cat for 3 days. His pay rate, x would be 12/3 = 4
He is paid $4 for feeding the cat per day.
To earn $40, the number of days that bill would have to work would be 40/4 = 10 days.
The neighbor's family is going on
vacation for 3 weeks next summer. There are 7 days in a week. Converting 3 weeks to days, it becomes 7 × 3 = 21 days.
The total amount of money that Bill will earn in 21 days would be
21 × 4 = $84
Since Bill wants to earn enough money to buy a CD player that costs $89, $84 won't be enough. He still needs $5 more and that would be from 2 more days.
A graphic distribution of the frequency and value of the numbers obtained while an imaging plate is being read is called what?
Answer:
Histogram
Step-by-step explanation:
A histogram is a type that has a wide application in the field of statistics. Histograms provide a visual interpretation of numerical data, indicating the number of data points within a range of values. These values are called classes or boxes. The frequency of data per class is illustrated by the use of a bar. The higher the rod, the higher the data values in the box. The following steps are followed to create a histogram
-Data of the group is sorted from small to large.
-The data group has an opening.
-The group width is calculated using the data opening and number of groups. The number of groups may be given to the question or asked to be determined by the solver.
The odd number closest to the number found is then taken as the group width. The reason for taking an odd number is to simplify the process by obtaining whole numbers in the calculations.
-The data is grouped in a group width and a table is created with the number of data belonging to each group.
-The groups in the table are placed on the vertical axis and the data numbers are placed on the horizontal axis and a histogram graph is created.
Mustafa, Heloise, and Gia have written more than a combined total of 222222 articles for the school newspaper. Heloise has written \dfrac{1}{4} 4 1 start fraction, 1, divided by, 4, end fraction as many articles as Mustafa has. Gia has written \dfrac{3}{2} 2 3 start fraction, 3, divided by, 2, end fraction as many articles as Mustafa has. Write an inequality to determine the number of articles, mmm, Mustafa could have written for the school newspaper.
Answer:
The Inequality For determining number of equation written by Mustafa for school paper is [tex]x+\frac{1}{4}x+ \frac{3}{2}x\geq 22[/tex].
Mustafa has written more than 8 articles.
Step-by-step explanation:
Given:
Combined Total Number of articles = 22
Let the number of articles written by Mustafa be 'x'.
Now Given:
Heloise has written [tex]\frac{1}{4}[/tex] as many articles as Mustafa has.
Number of article written by Heloise = [tex]\frac{1}{4}x[/tex]
Gia has written [tex]\frac{3}{2}[/tex] as many articles as Mustafa has.
Number of article written by Gia = [tex]\frac{3}{2}x[/tex]
Now we know that;
The sum of number of articles written by Mustafa and Number of article written by Heloise and Number of article written by Gia is greater than or equal to Combined Total Number of articles.
framing in equation form we get;
[tex]x+\frac{1}{4}x+ \frac{3}{2}x\geq 22[/tex]
Hence the Inequality For determining number of equation written by Mustafa for school paper is [tex]x+\frac{1}{4}x+ \frac{3}{2}x\geq 22[/tex].
Now Solving the Inequality we get;
Taking LCM for making the denominator common we get:
[tex]\frac{x\times 4}{4}+\frac{1\times1}{4\times1}x+ \frac{3\times2}{2\times2}x\geq 22\\\\\frac{4x}{4}+ \frac{x}{4}+\frac{6x}{4}\geq 22\\\\\frac{4x+x+6x}{4} \geq 22\\\\11x\geq 22\times4\\\\11x\geq 88\\\\x\geq \frac{88}{11} \\\\x\geq 8[/tex]
Hence Mustafa has written more than 8 articles.
Answer:
inequality - m+ 1/4m + 3/2m > 22
solution set - m>8
Step-by-step explanation:
Translate the following into an inequality:
Eight is less than twice what number?
8 < 2n
8 < 2 - n
n < 8 × 2
2 < 8n
Answer:
The right inequa is
8 < 2n
:)
Answer:
8<2n is correct :) Hope it helped
Write in simple radical form:
(1 +[tex]\frac{1}{\sqrt{3} }[/tex]) / (1 - [tex]\frac{1}{\sqrt{3} }[/tex])
show steps.
Answer:
The answer to your question is [tex]\frac{2}{3}[/tex]
Step-by-step explanation:
Here, there is a difference of squares, so just solve it and simplify
[tex]( 1 + \frac{1}{\sqrt{3} } ) (1 - \frac{1}{\sqrt{3} } )[/tex]
Multiply the binomials
= 1 - [tex]\frac{1}{3}[/tex]
Simplify
= [tex]\frac{3 - 1}{3}[/tex]
= [tex]\frac{2}{3}[/tex]
Georgina was given that the length of the rectangle was 2.5 inches longer than its width, and that the perimeter of the rectangle was 75.4 inches. Algebraically, find the length and width of the rectangle.
Answer: length = 20.1
Width=17.6
Step-by-step explanation: