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 $1349.
R, left parenthesis, x, right parenthesis, equals, 20, x
R(x)=
20x
The revenue function.
C, left parenthesis, x, right parenthesis, equals, x, plus, 1463
C(x)=
x+1463
The cost function.

Answer 1

To calculate the profit function \( P(x) \), we use the formula:

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

where \( R(x) \) is the revenue function and \( C(x) \) is the cost function.

Given:

\[
R(x) = 20x
\]

\[
C(x) = x + 1463
\]

Now, substituting \( R(x) \) and \( C(x) \) into the profit function:

\[
P(x) = 20x - (x + 1463)
\]

Simplifying the expression:

\[
P(x) = 20x - x - 1463
\]
\[
P(x) = 19x - 1463
\]

Now we need to determine how many units must be produced for the business to make a profit of $1349. We set the profit function equal to $1349:

\[
19x - 1463 = 1349
\]

Now we solve for \( x \):

1. Add 1463 to both sides:
\[
19x = 1349 + 1463
\]
\[
19x = 2812
\]

2. Divide by 19:
\[
x = \frac{2812}{19}
\]
\[
x = 148
\]

Therefore, the business must produce **148 units** to make a profit of $1349.