To solve the problem step by step:
Part 1: Profit Function
The profit \( P(x) \) is calculated as the difference between total revenue \( R(x) \) and total cost \( C(x) \).
Given:
- Cost function: \( C(x) = 15x + 3611 \)
- Revenue function: \( R(x) = 82x \)
The profit function can be expressed as: \[ P(x) = R(x) - C(x) \]
Substituting in the given functions: \[ P(x) = 82x - (15x + 3611) \] Simplifying this: \[ P(x) = 82x - 15x - 3611 \] \[ P(x) = 67x - 3611 \]
Part 2: Profit for 150 Bicycles
To find the profit for \( x = 150 \): \[ P(150) = 67(150) - 3611 \] Calculating this: \[ P(150) = 10050 - 3611 = 6439 \]
So, the profit is \( $6439 \).
Part 3: Average Profit per Bicycle Function
The average profit per bicycle, denoted as \( \overline{P}(x) \), is the total profit divided by the number of bicycles \( x \): \[ \overline{P}(x) = \frac{P(x)}{x} \]
Substituting the profit function we derived: \[ \overline{P}(x) = \frac{67x - 3611}{x} = 67 - \frac{3611}{x} \]
Part 4: Average Profit per Bicycle for 150 Bicycles
To find the average profit per bicycle when \( x = 150 \): \[ \overline{P}(150) = 67 - \frac{3611}{150} \] Calculating \( \frac{3611}{150} \): \[ \frac{3611}{150} \approx 24.0733 \] Therefore: \[ \overline{P}(150) = 67 - 24.0733 \approx 42.9267 \]
Rounding to the nearest cent: \[ \overline{P}(150) \approx 42.93 \]
Final Summary:
- Part 1: \( P(x) = 67x - 3611 \)
- Part 2: The profit is \( $6439 \)
- Part 3: \( \overline{P}(x) = 67 - \frac{3611}{x} \)
- Part 4: The average profit per bicycle is \( $42.93 \) (rounded to the nearest cent).