Question

Natalie is a hair stylist. When making appointments for her clients, she allots 30 minutes for

each cut and two hours for each cut and color. She has ten hours available for appointments each
day and prefers to see no more than 8 total clients. Each routine cut costs $40 and each cut and
color costs $100. How many of each type of appointment should she schedule each day to maxmize
her income?

Answer 1

To maximize her income, Natalie should schedule a combination of routine cuts and cuts with color appointments. We need to solve the constraints of available time and number of clients to find the feasible region and maximize the income function. Linear programming techniques can be used to solve this problem.

Step-by-step explanation:

To maximize her income, Natalie should schedule a combination of routine cuts and cuts with color appointments.

Let's assume Natalie schedules 'x' routine cuts and 'y' cuts with color.

1. Each routine cut takes 30 minutes, so the total time for routine cuts is 30x minutes.
2. Each cut with color takes 2 hours, so the total time for cuts with color is 2y hours.
3. Natalie has 10 hours available for appointments each day, so the total time for all appointments is equal to 30x + 120y minutes.
4. Natalie prefers to see no more than 8 total clients, so the total number of appointments is equal to x + y.
5. The income generated from routine cuts is $40x and the income generated from cuts with color is $100y.
6. Therefore, Natalie's income function is 40x + 100y.

To maximize her income, we need to solve the following constraints:

1. 30x + 120y ≤ 600 (10 hours = 600 minutes)
2. x + y ≤ 8 (no more than 8 clients)

We can graph these constraints on a graph and find the feasible region where x and y are both positive integers and find the maximum value of the income function.

Alternatively, we can also use linear programming techniques to solve this problem.