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### Writing and Using Linear Functions

Identify the relevant information given in the problem below and use it to answer the following questions. Round your answers to two decimal places as needed.

Gustavo got a job selling caramel apples at the local market. He is paid a small fixed amount plus a percent of each item he sells. The table below shows how much he gets paid depending on the number of items sold. The data is exactly linear.

| Items Sold (n) | 10 | 50 | 100 | 500 | 1000 | 5000 |
|----------------|------|------|------|------|------|------|
| Pay, P, in Dollars | 130.4 | 172 | 224 | 640 | 1160 | 5320 |

a) Identify the vertical intercept and average rate of change for the data.
- Vertical intercept = [tex]$120.00$[/tex]
- Average Rate of Change = [tex]$1.04$[/tex]

b) Use your results from part a) to write the linear function that represents the data table. Use the indicated variables and proper function notation.
[tex]\[ P(n) = 1.04n + 120.00 \][/tex]

Use your function to determine how much Gustavo will be paid if he sells 440 caramel apples.
[tex]\[ P(440) = 577.60 \][/tex]



Answer :

GinnyAnswer
Certainly! Let's go through the solution step-by-step.

### Given Information:
We have the following table showing how much Gustavo gets paid depending on the number of items sold:
[tex]\[ \begin{array}{|c|r|r|r|r|r|r|} \hline \text{Items Sold } (n) & 10 & 50 & 100 & 500 & 1000 & 5000 \\ \hline \text{Pay, } P, \text{ in Dollars} & 130.4 & 172 & 224 & 640 & 1160 & 5320 \\ \hline \end{array} \][/tex]
Given that the data is exactly linear, we can fit a linear equation of the form [tex]\( P(n) = \text{slope} \times n + \text{intercept} \)[/tex].

### Part A: Identify the Vertical Intercept and Average Rate of Change
To find the vertical intercept and the average rate of change (slope), we use linear regression principles where:
- The vertical intercept (or [tex]\( y \)[/tex]-intercept) is the value of [tex]\( P \)[/tex] when [tex]\( n = 0 \)[/tex].
- The average rate of change (slope) represents how much [tex]\( P \)[/tex] changes with respect to [tex]\( n \)[/tex].

Based on calculations from the given data, we find:
- The vertical intercept is [tex]\( \approx 120.00 \)[/tex].
- The average rate of change (slope) is [tex]\( \approx 1.04 \)[/tex].

So, we have:
- Vertical intercept, [tex]\( b \approx 120.00 \)[/tex]
- Average rate of change (slope), [tex]\( m \approx 1.04 \)[/tex].

### Part B: Write the Linear Function
Using the results from part (a), we can write the linear function in the form [tex]\( P(n) = m n + b \)[/tex]:

[tex]\[ P(n) = 1.04 n + 120.00 \][/tex]

### Part C: Determine Pay for Selling 440 Items
To determine how much Gustavo will be paid if he sells 440 caramel apples, we substitute [tex]\( n = 440 \)[/tex] into our linear function [tex]\( P(n) \)[/tex]:

[tex]\[ P(440) = 1.04 \times 440 + 120.00 \][/tex]

By performing this calculation, we get:

[tex]\[ P(440) = 457.60 + 120.00 = 577.60 \][/tex]

So, Gustavo will be paid [tex]\( \$577.60 \)[/tex] if he sells 440 caramel apples.

### Summary
- The vertical intercept is [tex]\( b \approx 120.00 \)[/tex].
- The average rate of change (slope) is [tex]\( m \approx 1.04 \)[/tex].
- The linear function is [tex]\( P(n) = 1.04 n + 120.00 \)[/tex].
- Using the function, the pay for selling 440 caramel apples is [tex]\( P(440) = 577.60 \)[/tex].

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