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 $1017.

Question

R, left parenthesis, x, right parenthesis, equals, 22, x
R(x)=
22x
The revenue function.
C, left parenthesis, x, right parenthesis, equals, 13, x, plus, 360
C(x)=
13x+360

Answer 1

To find the profit function, we start with the definitions of revenue \( R(x) \) and cost \( C(x) \). The profit function \( P(x) \) is defined as the revenue minus the cost:

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

Given: \[ R(x) = 22x \] \[ C(x) = 13x + 360 \]

Substituting these into the profit equation, we get:

\[ P(x) = 22x - (13x + 360) \]

Now simplify this expression:

\[ P(x) = 22x - 13x - 360 \] \[ P(x) = 9x - 360 \]

So the profit function is:

\[ P(x) = 9x - 360 \]

Now, to determine how many units must be produced for the business to make a profit of $1017, we can set the profit function equal to 1017:

\[ 9x - 360 = 1017 \]

Now, solve for \( x \):

1. Add 360 to both sides: \[ 9x = 1017 + 360 \] \[ 9x = 1377 \]

2. Divide both sides by 9: \[ x = \frac{1377}{9} \] \[ x = 153 \]

Therefore, the number of units that must be produced for the business to make a profit of $1017 is \( \boxed{153} \).