# Linear Programming Homework 10 questions

I’m studying and need help with a Graphs question to help me learn.

1. An objective function and a system of linear inequalities representing constraints are given. Complete parts (a) through (c).

Objective Function z=6x+7y

Constraints

 x+y≤9 x+2y≤10

a. Graph the system of inequalities representing the constraints on the given graph of the first Quadrant.

b. Find the value of the objective function at each corner of the graphed region bounded by the given constraints and and boundaries of the first quadrant x is greater than or equal to 0 and y is greater than or equal to 0

c. Find the maximum value and the coordinate of the maximum value.

2. A hurricane has just devastated an area, and relief teams are gathering supplies to take to that area. They are packing cases of food and water. Let f=number of cases of food and w=number of cases of water. If they can pack up to 400 cases on one truck, write an inequality that expresses this.

3. Assume the company makes a profit of \$15 per bookshelf and \$40 per desk. They must manufacture between 30 and 70 bookshelves per day, inclusive. They must also make between 25 and 65 desks per day, inclusive. Let x equal bookshelves and y represent desks. Write the inequalities for the two constraints.

4. Assume the company makes a profit of \$15 per bookshelf and \$40 per desk. They must manufacture between 10 and 80 bookshelves per day, inclusive. They must also make between 25 and 65 desks per day, inclusive. Let x equal bookshelves and y represent desks. To maintain high quality, they should not manufacture more than 90 pieces per day. How many bookshelves and how many desks should be manufactured per day to obtain a maximum profit?

5. A student earns \$15 per hour for tutoring and \$8 per hour as a teacher’s aide. Let x=the number of hours each week spent tutoring and y=the number of hours each week spent as a teacher’s aide. Complete parts (a) through (e).

a. Write the objective function that describes total weekly earnings.

b. The student is bound by three constraints. Write an inequality for each constraint.

c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because x and y are nonnegative.

d. Evaluate the objective function for total weekly earnings at each of the four vertices of the graphed region.

e. Complete the missing portions of the statement below.

The students can earn the maximum amount per week by tutoring for

hours per week and working as a teacher’s aide for

hours per week. The maximum amount that the student can earn each week is

\$

6. Food and clothing are shipped to victims of a natural disaster. Each carton of food will feed 12 people, while each carton of clothing will help 5 people. Each 25-cubic-foot box of food weighs 40 pounds and each 5-cubic-foot box of clothing weighs 25 pounds. The commercial carriers transporting food and clothing are bound by the following constraints: The total weight per carrier cannot exceed 23,000 pounds. The total volume must be no more than 8000 cubic feet. Use this information to answer the following questions. How many cartons of food and clothing should be sent with each plane shipment to maximize the number of people who can be helped?

7. During a war, allies sent food and medical kits to help survivors. Each food kit helped 8 people and each medicine kit helped 6 people. Each plane could carry no more than 50,000 pounds. Each food kit weighed 20 pounds and each medicine kit weighed 10 pounds. In addition to the weight constraint on its cargo, each plane could carry a total volume of supplies that did not exceed 4000 cubic feet. Each food kit was 1 cubic foot and each medical kit also had a volume of 1 cubic foot. Assume that those helped by medicine kits were not helped by the food kits and vice versa. What was the maximum number of people that could be helped with one plane of supplies?

8. Determine whether the following statement makes sense or does not make sense, and explain your reasoning.

In order to solve a linear programming problem, I use the graph representing the constraints and the graph of the objective function.

A.

The statement does not make sense because the graph of the objective function is not required.

The statement makes sense because the optimal solution is represented by the intersection of the two graphs.

The statement does not make sense because the graph representing the constraints is not required.

The statement makes sense because the graph of the objective function is used to find the corner points and the graph representing the constraints is used to determine which one is the best.

9. Determine whether the following statement makes sense or does not make sense, and explain your reasoning.

I use the coordinates of each vertex from my graph representing the constraints to find the values that maximize or minimize an objective function.

The statement makes sense because the coordinates of the vertices represent the values of the variables in the objective function.

The statement does not make sense because the maximum or minimum value of the objective function might not be at the vertices.

The statement makes sense because the coordinates of the vertices represent the value of the objective function.

The statement does not make sense because the graph of the objective function is used to find the vertices.

10. Determine whether the following statement makes sense or does not make sense, and explain your reasoning.

An important application of linear programming for businesses involves maximizing profit.