Question

Suppose the weekly cost for the production and sale of bicycles is ​C(x)equals15xplus3611 dollars and that the total revenue is given by ​R(x)equals82x ​dollars, where x is the number of bicycles.
a.
Write the equation of the function that models the weekly profit from the production and sale of x bicycles.
b.
What is the profit on the production and sale of 150 ​bicycles?
c.
Write the function that gives the average profit per bicycle.
d.
What is the average profit per bicycle if 150 are produced and​ sold?
Question content area bottom
Part 1
a.​ P(x)equals
  
enter your response here
Part 2
b. The profit is ​$
  
enter your response here.
Part 3
c. Upper P overbar left parenthesis x right parenthesis equals
  
enter your response here
Part 4
d. The average profit per bicycle is ​$
  
enter your response here. ​(Round to the nearest cent as​ needed.)

Answer 1

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).