Question

Jimmy owns a small engine repair business. The revenue, in dollars, can be modeled by the equation y = 420 + 72x, where x is the number of hours spent repairing small engines. The overhead cost, in dollars, can be modeled by the equation y = 24x² + 180 where x is the number of hours spent repairing bikes. After about how many hours does the company break even? Note: The phrase break even refers to the value where the two functions are equivalent.

Answer 1

5 hours

Explanation:

Given

`y = 420 + 72x` --- Revenue

`y = 24x^2 + 180` --- Overhead cost

Required

Determine the hours for break even

To do this, we simply equate both expressions as follows:

`24x^2 + 180 = 420 + 72x`

Collect Like Terms

`24x^2 - 72x + 180 - 420 = 0`

`24x^2 - 72x -240 = 0`

Divide through 24

`x^2 - 3x - 10 = 0`

Expand:

`x^2 -5x + 2x -10 = 0`

Factorize:

`x(x -5) + 2(x -5) = 0`

`(x + 2)(x -5) = 0`

`x + 2 = 0\ or\ x - 5 = 0`

`x = -2\ or\ x = 5`

But x can't be negative because it represents time.

So, x = 5 hours