Question

The revenue for a business, as a function of units produced, x, is shown below by R(x). C(x) represents the cost of producing x units. Calculate the profit function and then determine how many units must be produced for the business to break even.
R, left parenthesis, x, right parenthesis, equals, 16, x
R(x)=
16x
The revenue function.
C, left parenthesis, x, right parenthesis, equals, 3, x, plus, 767
C(x)=
3x+767
The cost function.

Answer 1

To calculate the profit function \(P(x)\), we use the formula:

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

Given the functions:

- Revenue function: \(R(x) = 16x\)
- Cost function: \(C(x) = 3x + 767\)

Substituting these into the profit equation:

\[
P(x) = R(x) - C(x) = 16x - (3x + 767)
\]

Simplifying:

\[
P(x) = 16x - 3x - 767
\]
\[
P(x) = 13x - 767
\]

Now, to find the break-even point, we set the profit function equal to zero and solve for \(x\):

\[
P(x) = 0 \implies 13x - 767 = 0
\]
\[
13x = 767
\]
\[
x = \frac{767}{13}
\]
\[
x = 59
\]

Thus, the business must produce **59 units** to break even.