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 spentrepairing small engines. the overhead cost in dollars can be modeled by the equation y=2x^2+180 where x is the number of hours spent repairing bikes. After about how many hours does the company break even?

Answer 1

The company breaks even at about 39 hours.

We will set both functions equal to each other, since we want to know when they break even:
420+72x = 2x²+180

We want to write this as a quadratic in standard form, so we will subtract 72x from both sides first:
420+72x-72x = 2x²+180-72x
420 = 2x²-72x+180

Now we subtract 420 from both sides:
420-420 = 2x²-72x+180-420
0 = 2x²-72x-240

We will now use the quadratic formula:
$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ \\=\frac{--72\pm \sqrt{(-72)^2-4(2)(-240)}}{2(2)} \\ \\=\frac{72\pm \sqrt{5184--1920}}{4} \\ \\=\frac{72\pm \sqrt{7104}}{4} \\ \\=\frac{72\pm 84.29}{4}=\frac{72+84.29}{4}\text{ or }\frac{72-84.29}{4} =\frac{-12.29}{4}\text{ or }\frac{156.29}{4} =-3.07\text{ or }39.07$

Since negative time makes no sense, the answer is 39.07, or about 39 hours.