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


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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?




Today's Homework

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Optimization Quiz


  • Click to see the exampleYou 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?
  • Click to see the exampleYou 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?
  • Click to see the exampleA 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?


  • Click to see the appletOptimization


  • Find the open box of largest volume that can be built from a 24" x 20" rectangle by removing squares from the corners.
    • click to see the videoSketch the rectangle to understand the problem
    • click to see the videoExperiment with different size boxes
    • click to see the videoThe 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?
    • click to see the videoVisualize the problem with different rectangles
    • click to see the videoThe solution using calculus

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