To solve the given problems, we'll follow each step accordingly.
a. Write the equation of the function that models the weekly profit from the production and sale of x bicycles.
The profit \( P(x) \) is calculated as the difference between the total revenue \( R(x) \) and the total cost \( C(x) \):
\[ P(x) = R(x) - C(x) \]
Given:
- \( C(x) = 30x + 4039 \)
- \( R(x) = 82x \)
Substituting the expressions for \( R(x) \) and \( C(x) \):
\[ P(x) = 82x - (30x + 4039) \]
Now simplifying that:
\[ P(x) = 82x - 30x - 4039 \] \[ P(x) = 52x - 4039 \]
b. What is the profit on the production and sale of 200 bicycles?
We substitute \( x = 200 \) into the profit function \( P(x) \):
\[ P(200) = 52(200) - 4039 \] \[ = 10400 - 4039 \] \[ = 6361 \]
So, the profit on the production and sale of 200 bicycles is $6361.
c. Write the function that gives the average profit per bicycle.
The average profit per bicycle can be represented as the total profit divided by the number of bicycles produced:
\[ \text{Average Profit} = \frac{P(x)}{x} \]
Substituting the profit function \( P(x) \):
\[ \text{Average Profit} = \frac{52x - 4039}{x} \]
Simplifying the expression gives:
\[ \text{Average Profit} = 52 - \frac{4039}{x} \]
d. What is the average profit per bicycle if 200 are produced and sold?
We substitute \( x = 200 \) into the average profit function:
\[ \text{Average Profit} = 52 - \frac{4039}{200} \] Calculating \( \frac{4039}{200} \):
\[ \frac{4039}{200} = 20.195 \]
Now substituting back:
\[ \text{Average Profit} = 52 - 20.195 \] \[ = 31.805 \]
So, the average profit per bicycle if 200 are produced and sold is approximately $31.81.