Question

Pham can work as many or as few hours as she wants at the college bookstore for \(12 per hour. But due to her hectic schedule, she has just 15 hours per week that she can spend working at either the bookstore or other potential jobs. One potential job at a café will pay her \)15 per hour for up to 6 hours per week. She has another job offer at a garage that will pay her \(13 an hour for up to 5 hours per week. And she has a potential job at a daycare center that will pay her \)11.50 per hour for as many hours as she can work. If her goal is to maximize the amount of money she can make each week, how many hours will she work at the bookstore?

Short answer

Pham will work for 4 hours at the bookstore.

Step-by-step solution

Estimating the income equation

Let the number of hours that Pham chooses to work at the daycare center be w, that at the bookstore be x, that at the café be y, and that at the garage be z. Her income equation will be

$M=11.5w+12x+15y+13z$

Income maximization

The daycare center will provide Pham with the least wages, which is $11.5 per hour for any number of hours she chooses to work. The marginal benefit in working at the bookstore rather than the daycare center is $0.5. Therefore, she will not work in the daycare center.

As the wages per hour at the café for 6 hours provide a maximum marginal benefit, she will work at the café for 6 hours. The next highest paying job is at the garage. So, she will work at the garage for 5 hours. She can work at the college bookstore for the remaining hours.

Job	Wages (in $ per hour)	Marginal Benefit (in $ per hour)
Café	15	2
Garage	13	1
Bookstore	12	0.5
Daycare center	11.50	-

Hours spent by Pham working in the bookstore

As Pham has a total of 15 hours to work, she will work at the café for 6 hours and at the garage for 5 hours. Then, she will work at the bookstore for the remaining hours. The hours can be estimated as follows.

$$\begin{gathered} 15=x+y+z \\ 15=x+6+5 \\ 15=x+11 \\ x=15-11=4 \end{gathered}$$

Therefore, Pham will work for 4 hours at the college bookstore.