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.

Question

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.