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, 23, x
R(x)=
23x
The revenue function.
C, left parenthesis, x, right parenthesis, equals, 12, x, plus, 902
C(x)=
12x+902
The cost function.

Answer 1

To calculate the profit function, we use the formula:

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

Given:
- The revenue function is \( R(x) = 23x \)
- The cost function is \( C(x) = 12x + 902 \)

Substituting these into the profit function:

\[
P(x) = R(x) - C(x) = 23x - (12x + 902)
\]

Now, simplify the profit function:

\[
P(x) = 23x - 12x - 902 = 11x - 902
\]

So, the profit function is:

\[
P(x) = 11x - 902
\]

Next, we determine the break-even point by setting the profit function \( P(x) \) to zero:

\[
0 = 11x - 902
\]

Solving for \( x \):

\[
11x = 902
\]
\[
x = \frac{902}{11}
\]
\[
x \approx 82
\]

The business must produce approximately **82 units** to break even.