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Choose the letter of the correct answer.

Tyler's car sales profits can be modeled using the function [tex]P(c)=c^2+4c[/tex], where [tex]c[/tex] is the number of cars and [tex]P(c)[/tex] is the sales (in thousands). What is the average rate of change in profit if he sells from 1 car to 5 cars?

A. [tex]\$2,000[/tex]
B. [tex]\$4,000[/tex]
C. [tex]\$6,000[/tex]
D. [tex]\$8,000[/tex]
E. [tex]\$10,000[/tex]



Answer :

GinnyAnswer
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]

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