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 make a profit of $738.
R, left parenthesis, x, right parenthesis, equals, 12, x
R(x)=
12x
The revenue function.
C, left parenthesis, x, right parenthesis, equals, 3, x, plus, 441
C(x)=
3x+441
The cost function.

Answer 1

To find the profit function \( P(x) \), we can use the following formula:

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

Given the revenue function \( R(x) = 12x \) and the cost function \( C(x) = 3x + 441 \), we can substitute these into the profit function:

\[
P(x) = R(x) - C(x) = 12x - (3x + 441)
\]

Now, simplify this expression:

\[
P(x) = 12x - 3x - 441 = 9x - 441
\]

So the profit function is:

\[
P(x) = 9x - 441
\]

Next, we need to determine how many units must be produced for the business to make a profit of \( 738 \). We set the profit function equal to \( 738 \):

\[
9x - 441 = 738
\]

Now, solve for \( x \):

1. Add \( 441 \) to both sides:
\[
9x = 738 + 441
\]
\[
9x = 1179
\]

2. Divide both sides by \( 9 \):
\[
x = \frac{1179}{9} = 131
\]

Therefore, the business must produce **131 units** to make a profit of **$738**.