Question
Suppose the weekly cost for the production and sale of bicycles is C(x)equals30xplus4039 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 200 bicycles?
c.
Write the function that gives the average profit per bicycle.
d.
What is the average profit per bicycle if 200 are produced and sold?
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 200 bicycles?
c.
Write the function that gives the average profit per bicycle.
d.
What is the average profit per bicycle if 200 are produced and sold?
Answers
GPT-4o mini
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**.
### 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**.