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Suppose the weekly cost for the production and sale of bicycles is ​C(x)equals24xplus3883 dollars and that the total revenue is given by ​R(x)equals81x ​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?
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Part 1
a.​ P(x)equals
  
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1 answer

To solve the problem, we’ll go through each part step by step.

Part a: Write the equation of the profit function

The profit function \( P(x) \) can be defined as the difference between total revenue \( R(x) \) and total cost \( C(x) \):

\[ P(x) = R(x) - C(x) \]

Given:

  • Cost function: \( C(x) = 24x + 3883 \)
  • Revenue function: \( R(x) = 81x \)

Substituting these into the profit function:

\[ P(x) = 81x - (24x + 3883) \] \[ P(x) = 81x - 24x - 3883 \] \[ P(x) = 57x - 3883 \]

Part b: Calculate the profit for 200 bicycles

Now we can calculate the profit when \( x = 200 \):

\[ P(200) = 57(200) - 3883 \] \[ P(200) = 11400 - 3883 \] \[ P(200) = 7517 \]

Part c: Write the function for average profit per bicycle

The average profit per bicycle can be calculated by dividing the total profit \( P(x) \) by the number of bicycles \( x \):

\[ \text{Average Profit per Bicycle} = \frac{P(x)}{x} = \frac{57x - 3883}{x} \]

Simplifying this,

\[ \text{Average Profit per Bicycle} = 57 - \frac{3883}{x} \]

Part d: Calculate the average profit per bicycle if 200 are produced and sold

We can now substitute \( x = 200 \) in the average profit function:

\[ \text{Average Profit per Bicycle} = 57 - \frac{3883}{200} \] \[ \text{Average Profit per Bicycle} = 57 - 19.415 \] \[ \text{Average Profit per Bicycle} \approx 37.585 \]

Summary of Answers

  • a. \( P(x) = 57x - 3883 \)
  • b. Profit on 200 bicycles: \( 7517 \) dollars
  • c. Average profit function: \( 57 - \frac{3883}{x} \)
  • d. Average profit per bicycle when 200 bicycles are produced: \( \approx 37.585 \) dollars