The figure shows the graphs of the cost and revenue functions for a company that manufactures and sells external hard drives. The solid line represents the revenue​ function, Upper R left parenthesis x right parenthesis equals 50 x​; the dashed line represents the cost​ function, Upper C left parenthesis x right parenthesis equals 10 comma 000 plus 30 x.

Question

a. Use the formulas to write the​ company's profit​ function, P, from producing and selling x hard drives.
b. Find the​ company's profit if 20 comma 000 hard drives are produced and sold.
Hard Drives Produced and Sold
0
200
400
600
800
0
10,000
20,000
30,000
40,000
x
y
Upper R left parenthesis x right parenthesis equals 50 xUpper C left parenthesis x right parenthesis equals 10 comma 000 plus 30 x

A coordinate plane has a horizontal x-axis labeled Hard Drives Produced and Sold from 0 to 800 in increments of 100 and a vertical y-axis labeled from 0 to 40000 in increments of 5000. A solid line labeled R (x) = 50 x rises from left to right from the origin and through the point (250, 12500). A dashed line labeled C (x) = 10,000 + 30 x rises from left to right from (0, 10000) and through the point (350, 20500). All coordinates are approximate.
Question content area bottom
Part 1
a. The profit a company makes is the money taken​ in, or the​ revenue, minus the money​ spent, or the cost.​ Therefore, if​ R(x) is the revenue and​ C(x) is the​ cost, the profit​ function, P, is represented as ​P(x)equals
Upper R left parenthesis x right parenthesis minus Upper C left parenthesis x right parenthesis .
Part 2
Substitute the equations for​ R(x) and​ C(x) into the profit function.
​P(x)
equals
Upper R left parenthesis x right parenthesis minus Upper C left parenthesis x right parenthesis
​P(x)
equals
  
enter your response hereminusleft parenthesis nothing right parenthesis

Answer 1

To find the company's profit function and calculate the profit for a specific number of hard drives produced and sold, we can proceed as follows:

Part 1: Profit Function

a. The profit function \( P(x) \) is defined as:

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

Given the revenue function \( R(x) = 50x \) and the cost function \( C(x) = 10,000 + 30x \):

Now, substituting these functions into the profit function:

\[ P(x) = 50x - (10,000 + 30x) \]

Simplifying, we combine like terms:

\[ P(x) = 50x - 10,000 - 30x \]

\[ P(x) = (50x - 30x) - 10,000 \]

\[ P(x) = 20x - 10,000 \]

Part 2: Calculate Profit for 20,000 Hard Drives

b. To find the profit when 20,000 hard drives are produced and sold, substitute \( x = 20,000 \) into the profit function:

\[ P(20,000) = 20(20,000) - 10,000 \]

Calculating this:

\[ P(20,000) = 400,000 - 10,000 \]

\[ P(20,000) = 390,000 \]

Summary of Results

1. The profit function is: \[ P(x) = 20x - 10,000 \]

2. The profit when 20,000 hard drives are produced and sold is: \[ P(20,000) = 390,000 \]