Linear Programming Practice

Chef Ben is going to make several chocolate cakes and sponge cakes, where each chocolate cakes needs 5 eggs and each sponge cake needs 7 eggs. Chef Ben plan to make not more than 7 chocolate cake and at least 3 sponge cakes. He can make a profit of $6 and $3 from selling each chocolate and each sponge cake respectively.

He wants to earn the maximum profit but there are only 70 eggs available.​

Given the objective function: P = 30x + 50y

Which vertex maximizes the profit?

The graph shows the constraints:

$$x\ +\ y\ \le\ 20$$

$$x\ \le\ 10$$

$$y\ \ge\ 6$$

It may be easier to see if you graph them on Desmos yourself. What are the critical points?

What do you call the area where all the shading overlaps?

A farmer can plant up to 6 acres of land with soybeans and corn. Her use of a necessary pesticide is limited by federal regulations to 15 gallons for her entire 6 acres. Soybeans require 2 gallons of pesticide for every acre planted and corn requires 3 gallons per acre. She will make $4000 profit per acre of soybeans and $3000 per acre of corn. If x represents acres of soybeans and y represents acres of corns, which inequalities represent the constraints to her situation?

A farmer can plant up to 6 acres of land with soybeans and corn. Her use of a necessary pesticide is limited by federal regulations to 15 gallons for her entire 6 acres. Soybeans require 2 gallons of pesticide for every acre planted and corn requires 3 gallons per acre. She will make $4000 profit per acre of soybeans and $3000 per acre of corn. If x represents acres of soybeans and y represents acres of corns, which expression represents the objective function to her situation?

A farmer can plant up to 6 acres of land with soybeans and corn. Her use of a necessary pesticide is limited by federal regulations to 15 gallons for her entire 6 acres. Soybeans require 2 gallons of pesticide for every acre planted and corn requires 3 gallons per acre. She will make $4000 profit per acre of soybeans and $3000 per acre of corn. If x represents acres of soybeans and y represents acres of corns. What are the critical points for her situation?

Maximize 3x + y with the following constraints:

$$x+y\ \le300$$ $$y\ \ge100$$   $$x\ \le150$$  

A doctor wishes to mix two types of foods in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 10 units of vitamin C. Food ‘x’ contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C. Food ‘y’ contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs $50 per kg to purchase Food ‘x’ and $70 per kg to purchase Food ‘y’. Formulate this problem as a linear programming problem to minimize the cost of such a mixture.

Which objective function and constraints fit this scenario?