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3.4. APPLIED OPTIMIZATION
• While there is no single algorithm that works in every situation where optimization
is used, in most of the problems we consider, the following steps are helpful: draw
a picture and introduce variables; identify the quantity to be optimized and find
relationships among the variables; determine a function of a single variable that models
the quantity to be optimized; decide the domain on which to consider the function
being optimized; use calculus to identify the absolute maximum and/or minimum of
the quantity being optimized.
Exercises
1. A rectangular box with a square bottom and closed top is to be made from two
materials. The material for the side costs $1.50 per square foot and the material for the
bottom costs $3.00 per square foot. If you are willing to spend $15 on the box, what is
the largest volume it can contain? Justify your answer completely using calculus.
2. A farmer wants to start raising cows, horses, goats, and sheep, and desires to have a
rectangular pasture for the animals to graze in. However, no two different kinds of
animals can graze together. In order to minimize the amount of fencing she will need,
she has decided to enclose a large rectangular area and then divide it into four equally
sized pens by adding three segments of fence inside the large rectangle that are parallel
to two existing sides. She has decided to purchase 7500 ft of fencing. What is the
maximum possible area that each of the four pens will enclose?
3. Two vertical poles of heights 60 ft and 80 ft stand on level ground, with their bases
100 ft apart. A cable that is stretched from the top of one pole to some point on the
ground between the poles, and then to the top of the other pole. What is the minimum
possible length of cable required? Justify your answer completely using calculus.
4. A company is designing propane tanks that are cylindrical with hemispherical ends.
Assume that the company wants tanks that will hold 1000 cubic feet of gas, and that the
ends are more expensive to make, costing $5 per square foot, while the cylindrical barrel
between the ends costs $2 per square foot. Use calculus to determine the minimum
cost to construct such a tank.
3.4. APPLIED OPTIMIZATION
• While there is no single algorithm that works in every situation where optimization
is used, in most of the problems we consider, the following steps are helpful: draw
a picture and introduce variables; identify the quantity to be optimized and find
relationships among the variables; determine a function of a single variable that models
the quantity to be optimized; decide the domain on which to consider the function
being optimized; use calculus to identify the absolute maximum and/or minimum of
the quantity being optimized.
Exercises
1. A rectangular box with a square bottom and closed top is to be made from two
materials. The material for the side costs $1.50 per square foot and the material for the
bottom costs $3.00 per square foot. If you are willing to spend $15 on the box, what is
the largest volume it can contain? Justify your answer completely using calculus.
2. A farmer wants to start raising cows, horses, goats, and sheep, and desires to have a
rectangular pasture for the animals to graze in. However, no two different kinds of
animals can graze together. In order to minimize the amount of fencing she will need,
she has decided to enclose a large rectangular area and then divide it into four equally
sized pens by adding three segments of fence inside the large rectangle that are parallel
to two existing sides. She has decided to purchase 7500 ft of fencing. What is the
maximum possible area that each of the four pens will enclose?
3. Two vertical poles of heights 60 ft and 80 ft stand on level ground, with their bases
100 ft apart. A cable that is stretched from the top of one pole to some point on the
ground between the poles, and then to the top of the other pole. What is the minimum
possible length of cable required? Justify your answer completely using calculus.
4. A company is designing propane tanks that are cylindrical with hemispherical ends.
Assume that the company wants tanks that will hold 1000 cubic feet of gas, and that the
ends are more expensive to make, costing $5 per square foot, while the cylindrical barrel
between the ends costs $2 per square foot. Use calculus to determine the minimum
cost to construct such a tank.