Question

MODELING WITH SYSTEMS OF INEQUALITIES

There are many situations that arise in business and engineering that necessitate systems of linear inequalities. The region in the xy-plane that solves the systems often represents all of the viable solutions to the system, so being able to visualize this region can be extremely helpful. As always, with modeling, it is important to really read the problems and understand the physical quantities involved.


(a) If John works 14 hours mowing and 30 hours at the bakery, does this satisfy all of the problem’s constraints?


(b) If x represents the hours John spends mowing and y represents the hours he spends at the bakery, write a system of inequalities that describes this scenario.

Answer 1

The question is incomplete. The complete question is :

John mows yards for his father's landscaping business for $ 10 per hour and also at a bakery for $ 15 per hour. He can work at most 52 hours per weak during the summer. He needs to make at least $ 600 per week to cover his living expenses.

a) If John works 14 hours mowing and 30 hours at the bakery, does this satisfy all of the problem’s constraints?

(b) If x represents the hours John spends mowing and y represents the hours he spends at the bakery, write a system of inequalities that describes this scenario.

Answer:

a). No

b). 10x + 15y ≥ 600

Explanation:

a). Number of hours John works mowing = 14 hours

Number of hours John works at bakery = 30 hours

Therefore, according tot he question,

= 14($10) + 30($15)

= 140 + 450

= $ 590

This is not enough money. He needs to earn $ 600 per week.

b). Given :

x = hours John spends mowing

y = hours John spends at the bakery

Therefore, the inequalities which describes this situation is:

x + y ≤ 52

10x + 15y ≥ 600