Applied Optimization Problems

A cylindrical can is to be made with a volume of 1000 cubic centimeters. Find the dimensions that will minimize the cost of materials used for the can.

• The dimensions that will minimize the cost of materials used for the can are r = (V / π)^(2/3) and h = (V / π)^(1/3).

• The dimensions that will minimize the cost of materials used for the can are r = (V / π)^(1/4) and h = (V / π)^(3/4).

• The dimensions that will minimize the cost of materials used for the can are r = (V / π)^(1/3) and h = (V / π)^(2/3).

• The dimensions that will minimize the cost of materials used for the can are r = (V / π)^(1/2) and h = (V / π)^(1/2).

A farmer wants to build a rectangular pen for his animals using an existing wall as one side. If he has 100 meters of fencing, what dimensions should he use to maximize the area of the pen?

• 50 meters

• 10 meters

• 75 meters

• 25 meters

A company wants to minimize the cost of producing a certain product. The cost function is given by C(x) = 1000 + 5x + 0.1x^2, where x is the number of units produced. Find the minimum cost and the corresponding production level.

• 0

• -25

• 1000

• 500

A rectangular box with a square base is to have a volume of 1000 cubic centimeters. Find the dimensions that will minimize the surface area of the box.

• 15

• 20

• 10

• 5

A company wants to maximize its profit. The profit function is given by P(x) = 500x - 0.1x^2, where x is the number of units sold. Find the maximum profit and the corresponding sales level.

• -500 units

• 1000 units

• 0 units

• 250 units

A cylindrical tank is to be made with a volume of 5000 cubic meters. Find the dimensions that will minimize the amount of material used for the tank.

• The dimensions that will minimize the amount of material used for the tank are a radius of 100 meters and a height of 50 meters.

• The dimensions that will minimize the amount of material used for the tank are a radius of √(10000 / π) meters and a height of 5000 / (π(√(10000 / π))^2) meters.

• The dimensions that will minimize the amount of material used for the tank are a radius of 500 meters and a height of 10 meters.

• The dimensions that will minimize the amount of material used for the tank are a radius of 10 meters and a height of 500 meters.

A rectangular garden is to be constructed using 40 meters of fencing. Find the dimensions of the garden that will maximize its area.

• 8 meters by 12 meters

• 10 meters by 10 meters

• 20 meters by 20 meters

• 5 meters by 15 meters

A company wants to minimize the cost of producing a certain product. The cost function is given by C(x) = 2000 + 10x + 0.2x^2, where x is the number of units produced. Find the minimum cost and the corresponding production level.

• -25

• 500

• 1000

• -2000

A closed rectangular shipping box with square base is to be made from 120 square inches of cardboard. What dimensions should the box be for maximum volume?

Choose the constraint and optimization equations that represent the problem. 

A square piece of green origami paper that is 6 inches on a side is being made into a gift box (with no lid) by cutting congruent squares out of each corner, folding up the sides, and taping the edges.

What size squares should you cut out for maximum volume? (do the whole problem)

• I should cut out squares that are 1/2 in by 1/2 in

• I should cut out squares that are 1 in by 1 in

• I should cut out squares that are 3 in by 3 in

• I should not cut out any squares