Calculus on Demand at Dartmouth College Lecture 28 | Index | Lecture 30 Lecture 29

## Resources

Math 3 Course Syllabus
Practice Exams

# Contents

In this lecture we learn to maximize or minimize a function on an interval by solving what are called "optimization problems."

### Quick Question

Among all the rectangles with perimeter equal to 4, which one has the largest area?

Optimization

### Quiz

Optimization Quiz

### Examples

• You want to run an underground power cable from a power station on one side of a river to a house on the other side. The house is 5 miles downstream from the station, and the river has a constant width of 1 mile. It costs \$1000 per mile to lay cable underground, and \$3000 per mile to lay cable under water. How should you lay the cable to minimize the total cost, and what will the minimum cost be?
• You want to smuggle a precious metal out of the country, by disguising it as a single cylindrical barrel, closed at both ends. The cost of shipping is \$7 per cubic foot. Once out the the country, you can sell the metal for \$8 per square foot. Assuming that you design the barrels with the height equal to twice the diameter, how many square feet should you smuggle, and what will your profit be?
• A wire 50 inches long is cut into two pieces. One piece is bent into a circle; the other, into a square. Where should the wire be cut to minimize the sum of the areas of the two shapes?

• Optimization

### Videos

• Find the open box of largest volume that can be built from a 24" x 20" rectangle by removing squares from the corners.
• Sketch the rectangle to understand the problem
• Experiment with different size boxes
• The solution using calculus

• Farmer Maria wishes to enclose the maximum area of pasture in a rectangle. One side of the pasture is already fenced. She has 100 feet of fence left. What dimensions should she make her field?
• Visualize the problem with different rectangles
• The solution using calculus

Lecture 28 | Index | Lecture 30