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 break even.
R, left parenthesis, x, right parenthesis, equals, 7, x
R(x)=
7x
The revenue function.
C, left parenthesis, x, right parenthesis, equals, 2, x, plus, 410
C(x)=
2x+410
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:

\[
R(x) = 7x
\]
\[
C(x) = 2x + 410
\]

Now, we can substitute these into the profit function:

\[
P(x) = 7x - (2x + 410)
\]

Simplifying this, we have:

\[
P(x) = 7x - 2x - 410
\]
\[
P(x) = 5x - 410
\]

### Break-even Point

To find the break-even point, we set the profit function equal to zero:

\[
P(x) = 0
\]

Substituting the profit function:

\[
5x - 410 = 0
\]

Now, solve for \( x \):

\[
5x = 410
\]
\[
x = \frac{410}{5}
\]
\[
x = 82
\]

### Conclusion

The profit function is given by:

\[
P(x) = 5x - 410
\]

The business must produce **82 units** to break even.