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 make a profit of $460.

Question

R, left parenthesis, x, right parenthesis, equals, 16, x
R(x)=
16x
The revenue function.
C, left parenthesis, x, right parenthesis, equals, 12, x, plus, 120
C(x)=
12x+120
The cost function.

Answer 1

To calculate the profit function, we subtract the cost function \( C(x) \) from the revenue function \( R(x) \). The profit function \( P(x) \) can be expressed as follows:

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

Substituting the expressions for \( R(x) \) and \( C(x) \):

\[ P(x) = 16x - (12x + 120) \]

Now, simplify \( P(x) \):

\[ P(x) = 16x - 12x - 120 \]

\[ P(x) = 4x - 120 \]

Now, we want to determine how many units must be produced for the business to make a profit of $460. We set \( P(x) \) equal to 460:

\[ 4x - 120 = 460 \]

Next, add 120 to both sides:

\[ 4x = 460 + 120 \]

\[ 4x = 580 \]

Now, divide by 4:

\[ x = \frac{580}{4} = 145 \]

Thus, the business must produce 145 units to make a profit of $460.