Answer :
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].
### 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].
