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, 30, x
R(x)=
30x
The revenue function.
C, left parenthesis, x, right parenthesis, equals, 4, x, plus, 2262
C(x)=
4x+2262
The cost function.

Answer 1

To calculate the profit function \( P(x) \), we start with the formulas for revenue \( R(x) \) and cost \( C(x) \):

1. The revenue function is given by:
\[
R(x) = 30x
\]

2. The cost function is given by:
\[
C(x) = 4x + 2262
\]

The profit function \( P(x) \) can be defined as the difference between revenue and cost:
\[
P(x) = R(x) - C(x)
\]

Substituting the revenue and cost functions into this equation gives:
\[
P(x) = 30x - (4x + 2262)
\]

Now, simplify the equation:
\[
P(x) = 30x - 4x - 2262
\]
\[
P(x) = 26x - 2262
\]

### Break-even Point
The break-even point occurs when the profit equals zero, meaning:
\[
P(x) = 0
\]

Setting the profit function to zero:
\[
26x - 2262 = 0
\]

Now, solve for \( x \):
\[
26x = 2262
\]
\[
x = \frac{2262}{26}
\]
\[
x = 87
\]

### Conclusion
The profit function is:
\[
P(x) = 26x - 2262
\]
The business must produce **87 units** to break even.