Question

he total revenue function for a product is given by Requals570x ​dollars, and the total cost function for this same product is given by Cequals18 comma 000plus80xplusxsquared​, where C is measured in dollars. For both​ functions, the input x is the number of units produced and sold.
a. Form the profit function for this product from the two given functions.
b. What is the profit when 34 units are produced and​ sold?
c. What is the profit when 48 units are produced and​ sold?
d. How many units must be sold to break even on this​ product?
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Question content area right
Part 1
a. Write the profit function.

Answer 1

To find the profit function, we first need to know the formulas for total revenue (R) and total cost (C):

1. **Total Revenue (R)**:
\[
R = 570x
\]

2. **Total Cost (C)**:
\[
C = 18,000 + 80x + x^2
\]

The profit function (P) is calculated using the following formula:
\[
P = R - C
\]

Substituting the expressions for revenue and cost into this formula gives us:

\[
P(x) = 570x - (18,000 + 80x + x^2)
\]

Now, simplify the profit function:

\[
P(x) = 570x - 18,000 - 80x - x^2
\]

Combining like terms:

\[
P(x) = (570x - 80x) - x^2 - 18,000
\]
\[
P(x) = 490x - x^2 - 18,000
\]

Thus, the profit function is:
\[
P(x) = -x^2 + 490x - 18,000
\]

Part 1: The profit function is:

\[
P(x) = -x^2 + 490x - 18,000
\]