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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.

Respuesta :

MrRoyal

Answer:

5 hours

Step-by-step explanation:

Given

[tex]y = 420 + 72x[/tex] --- Revenue

[tex]y = 24x^2 + 180[/tex] --- Overhead cost

Required

Determine the hours for break even

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

[tex]24x^2 + 180 = 420 + 72x[/tex]

Collect Like Terms

[tex]24x^2 - 72x + 180 - 420 = 0[/tex]

[tex]24x^2 - 72x -240 = 0[/tex]

Divide through 24

[tex]x^2 - 3x - 10 = 0[/tex]

Expand:

[tex]x^2 -5x + 2x -10 = 0[/tex]

Factorize:

[tex]x(x -5) + 2(x -5) = 0[/tex]

[tex](x + 2)(x -5) = 0[/tex]

[tex]x + 2 = 0\ or\ x - 5 = 0[/tex]

[tex]x = -2\ or\ x = 5[/tex]

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

So, x = 5 hours

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