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 $2376.
R, left parenthesis, x, right parenthesis, equals, 24, x
R(x)=
24x
The revenue function.
C, left parenthesis, x, right parenthesis, equals, 2, x, plus, 1364
C(x)=
2x+1364
The cost function.

Answer 1

To calculate the profit function \( P(x) \), we need to subtract the cost function \( C(x) \) from the revenue function \( R(x) \):

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

Given:

- \( R(x) = 24x \)
- \( C(x) = 2x + 1364 \)

Substituting these into the profit function:

\[
P(x) = 24x - (2x + 1364)
\]

Now, simplify the equation:

\[
P(x) = 24x - 2x - 1364
\]
\[
P(x) = 22x - 1364
\]

Now, we want to determine how many units \( x \) must be produced for the business to make a profit of $2376. We set \( P(x) = 2376 \):

\[
22x - 1364 = 2376
\]

To solve for \( x \), first, add 1364 to both sides:

\[
22x = 2376 + 1364
\]
\[
22x = 3740
\]

Now, divide both sides by 22:

\[
x = \frac{3740}{22}
\]
\[
x = 170
\]

Thus, the number of units that must be produced for the business to make a profit of $2376 is \( \boxed{170} \).