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​ company's profit function is Upper P left parenthesis x right parenthesis equals
  
390,000. ​(Simplify your​ answer.)

Answer 1

To find the profit function \( P(x) \), we first need to define the profit in relation to the revenue and cost functions.

a. Profit Function

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

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

Given the revenue function \( R(x) = 50x \) and the cost function \( C(x) = 10,000 + 30x \), we can substitute these into the profit function:

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

Simplifying this:

\[ P(x) = 50x - 10,000 - 30x \] \[ P(x) = (50x - 30x) - 10,000 \] \[ P(x) = 20x - 10,000 \]

b. Profit for 20,000 Hard Drives

Now we need to find the company's profit if 20,000 hard drives are produced and sold:

\[ P(20,000) = 20(20,000) - 10,000 \] Calculating this gives:

\[ P(20,000) = 400,000 - 10,000 \] \[ P(20,000) = 390,000 \]

Conclusion

The company's profit function is:

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

And the profit when 20,000 hard drives are produced and sold is:

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