The area of a square is decreasing at a rate of 43 square inches per second. At the time when the side length of the square is 7, what is the rate of change of the perimeter of the square? Round your answer to three decimal places (if necessary).

Each rope was 84.64 meters long.

Step-by-step explanation:

Given,

No. of ropes bought = 66

Total length = 5586.5 meters

Length of one rope = [tex]\frac{Total\ length}{No.\ of\ ropes\ bought}\\[/tex]

[tex]Length\ of\ one\ rope=\frac{5586.5}{66}\\Length\ of\ one\ rope=84.64\ meters[/tex]

Hence,

Each rope was 84.64 meters long.

Keywords: Division.

Learn more about division at:

#LearnwithBrainly

Answer:a is the correct option

Step-by-step explanation:

The table is divided into columns for month, sales and expense

Total number of sales for four months is determined by adding up the sales for each month. It becomes

35.75+65.34+12.15+49.68= $162.92

Total number of sales for four months is determined by adding up the expenses for each month. It becomes

43.18+52.24+41.09+59.50=196.01

Profit or loss = total sales - total expenses

Therefore 162.92-196.01 = - $33.09

It's a loss because it is negative

Answer:

The answer to your question is the last option

Step-by-step explanation:

Here there is a rational function so, we need to find the values where the denominator equals zero.

                         [tex]f(x) = \frac{x^{2} + 6x + 5}{x^{2}- 25 }[/tex]

                                [tex]x^{2} - 25 = 0[/tex]

Factor                      (x - 5)(x + 5) = 0

                               x₁ - 5 = 0               x₂ + 5 = 0

The function does not exist in -5 and 5

                              x₁ = 5                    x₂ = -5

Domain

                      (  -∞ , -5) U (-5, 5) U (5, ∞)

Answer:

the answer is d in edge

Step-by-step explanation:

(–∞, –5) U (–5, 5) U (5, ∞)

Answer:

A local café serves tea, coffee, cookies, scones, and muffins. They recently gathered data about their customers who purchase both a drink and a snack. The given frequency table shows the results of the survey.

If approximately 24% of the customers surveyed have a scone with their tea and approximately 36% of the customers surveyed buy a muffin, complete the column and row headings for the given table.

Step-by-step explanation:

you have a list of conversion factors.Just use them to cancel the unwanted units:246.6ft * 12in/1ft * 6lenthi/39in * 7ganthi/345.6lenthii= (246.6*12*6*7)/(1*39*345.6) ganthii

Answer:

Internal users refer to management members of a company and other people who use financial information to run and manage the business

Explanation:

They work within the business and make business decisions. Accounting information is important for all management levels. In order to make decisions about the company's future, the top managers need information about the past performance of the company.

To determine the results of their past decisions, they need data. The financial statements will determine the company's performance and its financial position and guide them in making future decisions.

Answer: A. To make business decisions and compare business performance with previous years

Explanation:

Management uses accounting information for evaluating and analyzing an organization's financial performance and position, to take important decisions and appropriate actions to improve the business performance in terms of profitability, financial position, and cash flows.

Internal users refer to the members of a company's management and other individuals who use financial information in running and managing the business. They work within the company and make decisions for the business.

Answer: $4.30

Step-by-step explanation: if you take 6.09 and subtract 1.79 you will get $4.30

Answer:

a. First derivative

Step-by-step explanation:

To determine the nature of the stationary points:

Either  Find  dy/dx or d2y/dx2

If the result is: positive—the point is a minimum one; negative—the point is a maximum one; zero—the point is a point of inflexion

or

Determine the sign of the gradient of the curve just before and just after the stationary points.

If the sign change for the gradient of the curve is: positive to negative—the point is a maximum one; negative to positive—the point is a minimum one; positive to positive or negative to negative— the point is a point of inflexion

A method for determining whether a critical point is a minimum, maximum, or neither using the slope of the curve is the First Derivative Test (Letter A).

Critical  points  represent the values of x for that f '(x) is zero or the derivative doesn't exist.  Therefore, they represent the only places where a function can have a local minimum, maximum or neither using the slope of the curve. See the rules for this method:

Read more about the First Derivative Test here:

https://brainly.com/question/6097697

Answer:

  v = 5cos(73°)i +5sin(73°)j

Step-by-step explanation:

The components of the velocity vector are ...

  horizontal (i direction): 5·cos(73°)

  vertical (j direction): 5·sin(73°)

Then the velocity vector is the sum of these component vectors:

  v = 5cos(73°)i +5sin(73°)j

Answer:

8x-y+4=0

Step-by-step explanation:

because 8x-y+4=-4+4

Answer:

Step-by-step explanation:

28 + 38 = 66

66 divided by 6 = 11

This means that there would be 11 pens in each box.

Answer: there were 12 red pens in each box

Step-by-step explanation:

Let x = the number of red pens that were initially in each box.

The college bookstore ordered six boxes of red pens.

This means that the total number of red pens ordered is 6×x = 6x

The store sold 28 red pens last week and 38 red pens this week. This means that the total number of red pens sold = 28 + 38 = 66

The number of red pens left will be the total number of red pens minus the number of red pens that were sold. It becomes 6x - 66

6 pens were left on the shelf. This means that

6x - 66 = 6

6x = 6 + 66 = 72

x = 72/6 = 12

Answer:

Step-by-step explanation:

A school has two computer labs and each lan has 30 computers. This means that the total number of computers is the sum of the number of computers in each lab

Total number of computers = 30+30 = 60

A total of six computers in the school or not working.

The expression that can be used to find the number of working computers in the school will be

Let the number of computers that are working be x

the number of computers that are working is total number of computers minus the number of computers that are not working be

The expression that can be used to find the number of working computers in the school will be

x = 60 - 6

x = 54

Answer:

[tex] log_{4}(2) = log_{ {2}^{2} }(2) = \frac{ log_{2}(2) }{2} = \frac{1}{2} [/tex]

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

Answer:

Option C.

Step-by-step explanation:

The given given sequence is

1, –3, 2, 5, –4, –6

We need to find the pairs of consecutive terms of the sequence and their product.

Pairs of consecutive terms | Product

     1, -3                                  -3

     -3, 2                                  -6

      2, 5                                  10

      5, -4                                -20

     -4, -6                                 24

Here the product of three pairs of consecutive terms is negative.

The number of variations in sign for the sequence is 3. Therefore, the correct option is C.

To find the number of sign variations in the sequence 1, -3, 2, 5, -4, -6, you count the changes from positive to negative or negative to positive between consecutive terms. The sequence has three such changes, so the answer is C. 3.

The question asks for the number of variations in sign within a given finite sequence of nonzero numbers. To find this, we need to count how many times the sign changes between consecutive numbers in the sequence 1, -3, 2, 5, -4, -6.

Counting these, we have a total of three variations in sign. Hence, the correct answer is C. 3.

Answer:

There are 672 seeds in 14 kg of apples

Step-by-step explanation:

2 kg of apples = 1 dozen

1 dozen = 12 apples

So, 2 kg of apples =12 apples

1 kg of apples = [tex]\frac{12}{2} apples[/tex]

1 kg of apples = [tex]6 apples[/tex]

14 kg of apples =[tex]6 \times 14=84 apples [/tex]

Each apples has 8 seeds

So, 84 apples has seeds = [tex]8 \times 84= 672 seeds[/tex]

Hence there are 672 seeds in 14 kg of apples

Apple seeds are in 14 kg of apples is mathematically given as

672 seeds

Question Parameters:

2 kg of apples is 1 dozen and that each apple has 8 seeds. How many apple seeds are in 14 kg of apples

Generally the equation for the   is mathematically given as

2 kg of apples = 1 dozen

1 dozen = 12 apples

Hence

1 kg of apples = 12/2

1 kg of apples = 6

14 kg of apples = 6*14

84 apples has seeds = 8*84

84 apples has seeds = 672

There are 672 seeds in 14 kg of apples

For more information on Arithmetic

https://brainly.com/question/22568180

Answer: a) Labor force participation rate = 72.46%, b) Unemployment rate = 1.8%.

Step-by-step explanation:

Since we have given that

Number of people who have at least a bachelor's degree = 71.9 millions

Number of people who were employed = 52.1 millions

Number of people who were unemployed = 1.3 millions

So, Labor- force participation rate would be

[tex]\dfrac{Employed}{Total}\times 100\\\\=\dfrac{52.1}{71.9}\times 100\\\\=72.46\%[/tex]

Unemployment rate for this group would be

[tex]\dfrac{Unemployed}{total}\times 100\\\\=\dfrac{1.3}{71.9}\times 100\\\\=1.8\%[/tex]

Hence, a) 72.46%, b) 1.8%.

The total number of elements in the union of the four sets is 195.

To find the total number of elements in the union of four sets, we need to subtract the number of elements shared by each pair of sets and the number of elements shared by each three of the sets. Here's the step-by-step calculation:

The total number of elements in the union of the four sets is 400 - 300 + 100 - 5 = 195 elements.

https://brainly.com/question/34747298

#SPJ3

The number of elements in the union of the four sets is 195.

To determine how many elements are in the union of four sets, we will use the principle of inclusion-exclusion. Let's denote the four sets as A, B, C, and D. According to the problem:

Each set has 100 elements:

|A| = |B| = |C| = |D| = 100

Each pair of sets shares 50 elements:

|A ∩ B| = |A ∩ C| = |A ∩ D| = |B ∩ C| = |B ∩ D| = |C ∩ D| = 50

Each three of the sets share 25 elements:

|A ∩ B ∩ C| = |A ∩ B ∩ D| = |A ∩ C ∩ D| = |B ∩ C ∩ D| = 25

All four sets share 5 elements: |A ∩ B ∩ C ∩ D| = 5

Using the principle of inclusion-exclusion for four sets, we have:

|A ∪ B ∪ C ∪ D| = |A| + |B| + |C| + |D| - (|A ∩ B| + |A ∩ C| + |A ∩ D| + |B ∩ C| + |B ∩ D| + |C ∩ D|) + (|A ∩ B ∩ C| + |A ∩ B ∩ D| + |A ∩ C ∩ D| + |B ∩ C ∩ D|) - |A ∩ B ∩ C ∩ D|

Substituting the given values:

|A ∪ B ∪ C ∪ D| = 100 + 100 + 100 + 100 - (50 + 50 + 50 + 50 + 50 + 50) + (25 + 25 + 25 + 25) -5

Combining these, we get:

|A ∪ B ∪ C ∪ D| = 400 - 300 + 100 - 5 = 195

So, the number of elements in the union of the four sets is 195.

Final answer:

The major difference between a calculator and a computer in performing calculations is that a calculator requires more human interaction for input and verification, whereas a computer can process complex tasks with less human assistance and is generally faster. Therefore, option A is the correct answer.

Explanation:

The primary difference between a calculator and a computer when performing calculations is that calculators typically require more human interaction to input and verify data, while computers can process complex tasks with less human assistance. A calculator is designed for straightforward numerical computations and requires accurate input from the user. Meanwhile, a computer can carry out a wide range of tasks, including calculations, with greater speed and autonomy, thanks to its ability to run complex software programs. However, even though computers are powerful, solving certain mathematical problems, such as differential equations, can be challenging since they cannot make infinitely fine time steps and must approximate using small ones. Moreover, calculators, being simpler devices, require very little energy to operate and often rely on solar cells or small batteries, unlike computers, which have a greater energy demand. Therefore, option A is the correct answer.

Answer:

Step-by-step explanation:

An ellipse is of the form:

[tex]\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1[/tex] if it is horizontally stretched, or

[tex]\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1[/tex] if it is vertically stretched.

To begin writing the equation of the ellipse, we will group together all the x-terms and then all the y-terms, moving the constant to the other side of the equals sign in order to make completing the square on these 2 terms a bit easier.

[tex](36x^2+360x)+(25y^2+100y)=-100[/tex]

The first rule for completing the square is that the leading coefficients on the squared terms HAVE to be a 1.  Neither of ours is, so we factor those out, getting:

[tex]36(x^2+10x)+25(y^2+4y)=-100[/tex]

Now we can complete the squares.  We will start with the x-term.  The rule is to take half the linear term, square it, then add it to both sides.  Our linear x-term is a 10.  Half of 10 is 5, and 5 squared is 25.  So we add 25 inside the parenthesis with the x stuff, BUT we cannot forget about that 36 hanging around out front refusing to be ignored.  It is a multiplier.  That means that we didn't just add in 25, we added in 25 * 36 which is 900.  So what we have right now is

[tex]36(x^2+10x+25)+25(y^2+4y)=-100+900[/tex]

Now we will do the same with the y-term.  Take half the linear term, square it, and add it (along with its multiplier) to both sides.  Our linear y-term is 4.  Half of 4 is 2, and 2 squared is 4, so now we have:

[tex]36(x^2+10x+25)+25(y^2+4y+4)=-100+900+100[/tex]

Now we will do 2 things simultaneously.  We will simplify the right side which is easy, and then create the perfect square binomials on the left that was the whole point of doing all that.  It now looks like this:

[tex]36(x+5)^2+25(y+2)^2=900[/tex]

Almost there.  Last thing we do is divide everything by 900 so we get a 1 on the right:

[tex]\frac{36(x+5)^2}{900}+\frac{25(y+2)^2}{900}=1[/tex]

Simplifying by division in both the rational terms:

[tex]\frac{(x+5)^2}{25}+\frac{(y+2)^2}{36}=1[/tex]

That means that this is a vertically stretched ellipse with a = 6 and b = 5.  In an ellipse, the a value is always the bigger one.  So if the bigger value in the denominator is under the x fraction, then it is a horizontal ellipse.  If the bigger value in the denominator is under the y fraction, like ours, then it is a vertical ellipse.

Answer:

Step-by-step explanation:

At 4 P.M., the total snowfall is 4 centimeters

Snowfall at 4 P.M = 4 cm

At 7 P.M., the total snow fall is 11 centimeters.

Snowfall at 7 P.M = 11 cm

Duration  = 7 - 4 = 3 hours

Snowfall = 11 - 4 = 7 cm

[tex]\texttt{Mean hourly snowfall = }\frac{\texttt{Snowfall}}{\texttt{Duration}}=\frac{7}{3}cm/hr[/tex]

Mean hourly snowfall = [tex]\frac{7}{3}[/tex] cm/hr

The equation for the situation is [tex]y=200(\frac{2}{3})^{x}[/tex]

It will take around 9 years for her to have around 5 sheep

Step-by-step explanation:

The exponential decay growth/decay equation is [tex]y=a(b)^{x}[/tex] , where

Erica is a sheep farmer. She is having a problem with wolves attacking the flock. She starts with an initial 200 sheep and notices that the population is two-thirds of the previous year

∵ The population is decreased

∴ The equation is decay

∵ The population is two-thirds of the previous year

∴ The decay factor is [tex]\frac{2}{3}[/tex]

∵ She starts with an initial 200 sheep

∴ the initial value is 200

∵ The decay equation is [tex]y=a(b)^{x}[/tex] , where y represent the

   population in x years

∵ a = 200

∵ b = [tex]\frac{2}{3}[/tex]

∴ [tex]y=200(\frac{2}{3})^{x}[/tex]

The equation for the situation is [tex]y=200(\frac{2}{3})^{x}[/tex]

∵ The population after x years is around 5 sheep

- Substitute y by 5 to find x

∵ [tex]5=200(\frac{2}{3})^{x}[/tex]

- Divide both sides by 200

∴ [tex]0.025=(\frac{2}{3})^{x}[/tex]

- Insert ㏒ for both sides

∴ [tex]log(0.025)=log(\frac{2}{3})^{x}[/tex]

∴ [tex]log(0.025)=xlog(\frac{2}{3})[/tex]

- Divide both sides by [tex]log(\frac{2}{3})[/tex]

∴ 9.098 = x

∴ x is around 9 years

It will take around 9 years for her to have around 5 sheep

Learn more:

You can learn more about the equation  in brainly.com/question/10666510

#LearnwithBrainly

Answer:

If the cost X is higher or equal than 35, then no one wants to buy the any mouse pads.

Step-by-step explanation:

When N is 0 or lower then it means that no one buys it.

N=35x-[tex]x^{2}[/tex], if N=0 then

0=35x-[tex]x^{2}[/tex] or

35x=[tex]x^{2}[/tex] dividing each side by x we get:

35=x

This is the cost which nobody will buy the mouse pad anymore.

Answer:

oooooooooo yea

Step-by-step explanation:

Answer:

  7 1/2 minutes

Step-by-step explanation:

When the time with both faucets is a submultiple of the time with one faucet, you can use this sort of reasoning to figure the missing time.

Filling the tub in 5 minutes is equivalent to having 3 faucets open, each of which can fill the tub in 15 minutes. Since we know the cold water faucet can fill the tub in 15 minutes, the hot water faucet is equivalent to two open 15-minute faucets. That is, it can fill the tub in 15/2 = 7 1/2 minutes.

_____

Perhaps a more conventional way to approach the problem is to consider "tubs per minute." The cold faucet runs at 1/15 tubs per minute, and the two faucets together run at 1/5 tub per minute. Then the hot runs at ...

  (1/5) - (1/15) = (3 -1)/15 = 2/15 . . . tubs per minute

So it will fill 2 tubs in 15 minutes, or 1 tub in 7 1/2 minutes.

[tex] \frac{d}{dx} {x}^{2} = 2x \\ x = 0 \\ 2 \times 0 = 0[/tex]

When the derivative equals zero, that point is either the maximum or minimum.

[tex] {x}^{2} \geqslant 0[/tex]

The smallest real value of x² is 0, which is at x = 0.

Hence, x² at x = 0 has both a local and global minimum.

Step-by-step explanation:

y = x²

Find the first derivative:

dy/dx = 2x

Find the critical values by setting dy/dx to 0:

0 = 2x

x = 0

Evaluating the first derivative before and after x=0, we see that dy/dx changes signs from negative to positive.

x < 0, dy/dx < 0

x > 0, dy/dx > 0

That means x=0 is a local minimum.

To find the global minimum, we need evaluate the function at x=0 and at the end points.

x = -∞, y = ∞

x = 0, y = 0

x = ∞, y = ∞

So x=0 is also the global minimum.

Answer:

Option A) It is a constant.

Step-by-step explanation:

We are given the following information in the question:

The general form of regression equation is:

[tex]Y = a + bX[/tex]

[tex]Y-bX = a[/tex]

where Y is the dependent variable and we want to predict the value of Y and X is the independent variable and the predictor.

If we compare the above equation with the point slope form:

[tex]y = mx + c[/tex]

where m is the slope of the line and c is the y-intercept.

We get,

Slope, m  = b

Y-intercept = a

Thus, a is the y-intercept that is the value of Y when X is 0.

It is a constant.

It can also be interpreted as the difference of Y and X when slope is 1.

The constant 'a' in the regression equation represents the y-intercept, which is the value of Y when X is equal to 0.

The constant a in the regression equation Y = a + bX is called the y-intercept. It represents the value of Y when X is equal to 0. In other words, it is the point where the regression line crosses the y-axis.

https://brainly.com/question/36829492

#SPJ3

Answer:

The answer is 1.5

Step-by-step explanation:

You have to find the pattern and work your way down to the 11th term.

Answer:

1.5

Step-by-step explanation:

The sequence given is a geometric progression (G.P). For a G.P where the first term is a, the common ratio (which is the ratio between successive terms), the nth term of the GP is given as

Tn = ar^n-1

From the sequence given  

a = 1536

r = 768/1536

= 1/2

= 0.5

Hence the 11th term of the sequence

T11 = 1536 (1/2)^11-1

= 1536 (1/2)^10

= 1536/1024

= 1.5

The degrees of freedom (df) value for a chi-square test for independence is calculated using the formula: df = (number of columns - 1) x (number of rows - 1).

The degrees of freedom (df) value for a chi-square test for independence when both variables are classified into two categories can be calculated using the formula:

df = (number of columns - 1) x (number of rows - 1)

For example, if we have a contingency table with 2 columns and 2 rows, the df value would be:

df = (2 - 1) x (2 - 1) = 1 x 1 = 1

Answer:

153 times

Step-by-step explanation:

We have to flip the coin in order to obtain a 95.8% confidence interval of width of at most .14

Width = 0.14

ME = [tex]\frac{width}{2}[/tex]

ME = [tex]\frac{0.14}{2}[/tex]

ME = [tex]0.07[/tex]

[tex]ME\geq z \times \sqrt{\frac{\widecap{p}(1-\widecap{p})}{n}}[/tex]

use p = 0.5

z at 95.8% is 1.727(using calculator)

[tex]0.07 \geq 1.727 \times \sqrt{\frac{0.5(1-0.5)}{n}}[/tex]

[tex]\frac{0.07}{1.727}\geq sqrt{\frac{0.5(1-0.5)}{n}}[/tex]

[tex](\frac{0.07}{1.727})^2 \geq \frac{0.5(1-0.5)}{n}[/tex]

[tex]n \geq \frac{0.5(1-0.5)}{(\frac{0.07}{1.727})^2}[/tex]

[tex]n \geq 152.169[/tex]

So, Option B is true

Hence  we have to flip 153 times the coin in order to obtain a 95.8% confidence interval of width of at most .14 for the probability of flipping a head

To get a 95.8% confidence interval of width 0.14 for the probability of flipping a head with a fair coin, we need to flip the coin approximately 213 times. This is achieved by setting the standard error of the confidence interval to half the desired width, and solving for n, the number of flips.

To answer the question, we need to use the formula for the confidence interval for a proportion, which uses the Z-score, the standard deviation of the distribution of sample proportions, and the desired width of the confidence interval.

In this case, we want a 95.8% confidence interval. The Z-score for a 95.8% confidence interval is approximately 2.054 (since the z-score was mentioned to be rounded to 3 decimal places in the question).

The standard deviation of the distribution of sample proportions (standard error, SE) is sqrt([P*(1-P)]/n), where P is the assumed probability of heads (0.5 for a fair coin), and n is the number of flips.

The desired width of the confidence interval is 0.14, so the maximum standard error we want is 0.14/2, or 0.07. Setting the standard error formula equal to 0.07 and solving for n, we get n = 0.5*(1-0.5)/(0.07/2.054)^2, which comes out to approximately 212.6.

However, since we can't have a partial flip of a coin, we round this up to the next whole number, 213, which is close to the choice (c) 212. Hence, the answer should be (f), None of the above.

https://brainly.com/question/35443908

#SPJ11

Answer:

a)1/4

b)1/2

Step-by-step explanation:

a) The selection is with replacement, then we already have 1 $ bill and in the hat again the four bills. What is the probability  of select 1$ bill ? well it is 1/4.

The point here is,  replacement condition make the second trial totally independent of the first one

b)Following the same reasoning now we have selected a 10 $ bill, and we have two possibles positive outcome a 10 $ and a 20 $ bill and a total of 4 outcomes th th probability is 1/2

The probabilities are:

a) 1/16

b) 1/8.

We assume that all the bills have the same probability of being randomly selected.

a) Then the probability of randomly selecting a $1 bill is equal to the number of $1 bills on the hat divided by the total number of bills.

There is one $1 bill on the hat, and 4 bills in total, so the probability is:

p = 1/4.

Then you return the $1 bill to the hat and want to draw it again, with the same probability as before.

q = 1/4.

The joint probability is the product of the individual probabilities, so we have:

P = p*q = (1/4)*(1/4) = 1/16.

b) There are 2 bills that have a larger value than $5, then the probability for the first draw is:

p = 2/4

For the second draw, we must get the $10 bill, and the probability of getting it is:

q = 1/4.

The joint probability is:

P = p*q = (2/4)*(1/4) = 1/8.

If you want to learn more about probability, you can read:

https://brainly.com/question/251701

Answer:

5/7 > 1/13. Hence Greater than Symbol will be used to compare the fractions.

Step-by-step explanation:

Given:

Two fractions given are 5/7 and 1/13

We will first calculate each of the fraction and then we will decide which one is greater and which one is smaller accordingly we will assign the symbol for the same.

First we will find the answer for 5/7.

Now dividing the number 5 from 7 we get;

[tex]5/7= 0.714[/tex]

Now we will find the answer for 1/13

Now dividing the number 1 from 13 we get;

[tex]\frac{1}{13}= 0.076[/tex]

From above we can see that 0.714 is greater than 0.076.

Hence we can say that 5/7 is greater than 1/13.

So the symbol used will be greater than > where number 5/7 is greater than 1/13

5/7 > 1/13

5/7>1/13 is the answer because 5/7 is closer to its whole when a number is closer to its whole its bigger for example 4/5>7/10 hope it helps

f(x) = 0.5x   -   money per ticket

g(x) = 8x   -   tickets per hour

We replace g(x) with its value (8x).

f(g(x)) = f(8x)

We replace the x in f(x) with 8x.

f(8x) = 0,5 × 8x

f(8x) = 4x

⇒ f(g(x)) = 4x

Correct answer:

a. f(g(x)) = 4x, which represents the money Aurora made in dollars per hour

Composing functions means to use the output of the inner function as the input for the outer function.

In this case, the inner function is g(x)=8x, which means that Aurora sells 8 tickets per hour.

So, if we give 8x (number of tickets sold) as input to the function f(x)=0.5x, we'll get

[tex]f(x)=0.5x \implies f(8x)=0.5(8x)=4x[/tex]

Since the function f tells us how much Aurora gains per ticket, f(g(x)) basically means:

Aurora gains 0.5 dollars per ticket, and she sells 8 ticket per hour, so she gains 4 dollars per hour.