Question

Kevin is an auto mechanic. He spends 3 hours when he replaces the shocks on a car and 2 hours when he replaces the brakes. He works no more than 42 hours a week. He routinely completes at least 4 shocks replacements and 6 brake replacements a week. If he charges $400 for labor replacing shocks and $200 in labor for replacing brakes, how many jobs of each type should he complete a week to maximize his income?

Answer 1

Let \(x\) be the number of shock replacements and \(y\) be the number of brake replacements per week. The objective is to maximize Kevin's income, which can be expressed as:

\[ I = 400x + 200y \]

Subject to the following constraints:

1. Time constraint: \(3x + 2y \leq 42\) (since he works no more than 42 hours a week)

2. Job completion constraints: \(x \geq 4\) (at least 4 shocks) and \(y \geq 6\) (at least 6 brakes)

Now, you can use these constraints to find the values of \(x\) and \(y\) that maximize (l).