Below are problems that have appeared (in one guise or another) in virtually every calculus textbook written in the past fifty years. Use Geometer's Sketchpad to construct a dynamic sketch, and then use it to make a conjecture about the problem and its solution.
1. Find the maximum area of an isosceles triangle whose perimeter is 18 inches.
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a. Write the area as a function of the length of the
side that is adjacent to the equal sides. b. Write the maximum area as a function of the perimeter. c. Show that for any triangle T there is an isosceles triangle whose area is the same as that of T and whose perimeter is no more than that of T. d. What does problem c imply? |
2. Your job is to plan the shape of an open-topped run-off gutter that will be imbedded in a dirt street to carry away rain water. The gutter is to be formed from a flat piece of metal that is 10 feet long and 8 inches wide. Make the gutter by bending up sides at right angles. Determine how long a side to bend up so that the gutter will carry as much water as possible.
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a. Write the cross-sectional area of the gutter as
a function of the length of the side of the gutter. b. Write the maximum area as a function of the width. c. If, instead of bending the metal at right angles, you could use any angle, could you do any better? That is, could a gutter of greater carrying capacity be made by bending up sides at less than a right angle? What about a "V-shaped" gutter? |
3. A triangular play area for a child is to be enclosed by two walls of a room and a straight security gate that will be attached to the walls. If the security gate is 6 feet long, at what points should it be attached to the walls to make the largest play area?
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a. Write the play area as a function of the length of one sides of the play area. i. When the length of the security gate is 6 feet b. Write the maximum area as a function of the length of the security gate. c. Write the play area as a function one of the angles of the triangular play area. |
4. What should the shape of a rectangular room of given perimeter, say 48 feet, be if the room is to have a maximum floor space?
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a. Write the floor area as a function of the length
of one side of the room. b. Write the maximum floor area as a function of the perimeter. c. What if, instead of a fixed perimeter and a maximum floor space, you want a fixed floor space, say 144 square feet, and for this fixed floor space you want a maximum wall space? d. Consider the initial problem of having a fixed perimeter of 48 feet and a desire to maximize the floor space. But this time instead of allowing only rectangular rooms, allow a pentagonal-shaped room. Will this shape allow more floor space given the fixed perimeter? What about a hexagon? |
5. For an agricultural experiment, a rectangular field is to be made by enclosing it by fence and then dividing it into three lots by fencing parallel to one of the sides. Find the dimensions of the largest field that can be enclosed with 800 feet of fencing.
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a. Write area of the field as a function of the length of one side of the field. i. When there is 48 feet of fencing b. Write the maximum field area as a function of the number of feet of fencing. c. What if, instead of three lots, you wanted just two lots within the large field? What if you needed four lots? n lots? |