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.

Question

R, left parenthesis, x, right parenthesis, equals, 30, x
R(x)=
30x
The revenue function.
C, left parenthesis, x, right parenthesis, equals, 4, x, plus, 2262
C(x)=
4x+2262
The cost function.

Answer 1

To calculate the profit function \( P(x) \), we start with the formulas for revenue \( R(x) \) and cost \( C(x) \):

1. The revenue function is given by: \[ R(x) = 30x \]

2. The cost function is given by: \[ C(x) = 4x + 2262 \]

The profit function \( P(x) \) can be defined as the difference between revenue and cost: \[ P(x) = R(x) - C(x) \]

Substituting the revenue and cost functions into this equation gives: \[ P(x) = 30x - (4x + 2262) \]

Now, simplify the equation: \[ P(x) = 30x - 4x - 2262 \] \[ P(x) = 26x - 2262 \]

Break-even Point

The break-even point occurs when the profit equals zero, meaning: \[ P(x) = 0 \]

Setting the profit function to zero: \[ 26x - 2262 = 0 \]

Now, solve for \( x \): \[ 26x = 2262 \] \[ x = \frac{2262}{26} \] \[ x = 87 \]

Conclusion

The profit function is: \[ P(x) = 26x - 2262 \] The business must produce 87 units to break even.