To solve this problem, we need to determine the average rate of change in profit based on the given profit function [tex]\( P(c) = c^2 + 4c \)[/tex] over the interval where Tyler sells from 1 car to 5 cars.
1. Calculate the profit for selling 1 car:
[tex]\[
P(1) = 1^2 + 4 \cdot 1 = 1 + 4 = 5 \text{ (in thousands of dollars)}
\][/tex]
2. Calculate the profit for selling 5 cars:
[tex]\[
P(5) = 5^2 + 4 \cdot 5 = 25 + 20 = 45 \text{ (in thousands of dollars)}
\][/tex]
3. Determine the average rate of change in profit:
The formula for the average rate of change of a function [tex]\( P(c) \)[/tex] from [tex]\( c = a \)[/tex] to [tex]\( c = b \)[/tex] is:
[tex]\[
\text{Average Rate of Change} = \frac{P(b) - P(a)}{b - a}
\][/tex]
Plugging in the values [tex]\( a = 1 \)[/tex] and [tex]\( b = 5 \)[/tex]:
[tex]\[
\text{Average Rate of Change} = \frac{P(5) - P(1)}{5 - 1} = \frac{45 - 5}{4} = \frac{40}{4} = 10 \text{ (in thousands of dollars per car)}
\][/tex]
4. Convert the average rate of change to dollars:
Since the profit [tex]\( P(c) \)[/tex] is given in thousands of dollars, the average rate of change is [tex]\( 10 \times 1000 = 10000 \)[/tex] dollars per car.
Thus, the average rate of change in profit if Tyler sells from 1 car up to 5 cars is:
[tex]\[
\boxed{10,000 \text{ dollars}}
\][/tex]
Given the multiple-choice options, the correct answer is:
[tex]\[
\text{E. } \$ 10,000
\][/tex]
