Question

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, ( r(x) = 50x ), the dashed line represents the cost function, ( c(x) = 10,000 + 30x )
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 25,000 hard drives are produced and sold

a. the company’s profit function is ( p(x) = square ) (simplify your answer.)

Explanation:

Part a

Step1: Recall Profit Formula

Profit \( P(x) \) is revenue minus cost, so \( P(x) = R(x) - C(x) \).

Step2: Substitute Functions

Given \( R(x) = 50x \) and \( C(x) = 10000 + 30x \), substitute into the profit formula:
\( P(x) = 50x - (10000 + 30x) \)

Step3: Simplify Expression

Distribute the negative sign: \( P(x) = 50x - 10000 - 30x \).
Combine like terms: \( P(x) = (50x - 30x) - 10000 = 20x - 10000 \).

Step1: Use Profit Function

We use the profit function \( P(x) = 20x - 10000 \) from part (a).

Step2: Substitute \( x = 23000 \)

Substitute \( x = 23000 \) into \( P(x) \):
\( P(23000) = 20(23000) - 10000 \)

Step3: Calculate Result

First, multiply \( 20 \times 23000 = 460000 \).
Then subtract 10000: \( 460000 - 10000 = 450000 \).

Answer:

\( P(x) = 20x - 10000 \)

Part b