Optimization Problems

1. David wishes to use 100 feet of fencing to enclose a rectangular garden. Determine the maximum possible area of his garden. ANS: 625 ft2.
2. Suppose David decides to place on sdie of the garden next to his house (in other words, he will not need fencing along the side of the house). Determine the maximum possible area of his garden. ANS: 1250 ft2.
3. Find the maximum volume of a box created by cuting squares from each corner of a 8 x 10 inch rectangle and folding up the sides. ANS: approx. 52.5 in3.
Here is an applet describing this situation:
http://www.cut-the-knot.org/Curriculum/Calculus/BoxVolume.shtml
4. A closed cylindrical can must have a volume of 1000 cubic inches. What dimensions will minimize its surface area? ANS: r approx. 5.4 in, h approx. 10.8 in.
5. Find the coordinates of the point on the curve x + y2 = 4 that is closest to the point (4, –4). ANS: approx. (2.72722 , –1.12817).
6. Postal regulations require that the sum of the length and girth of a rectangular package may not exceed 108 inches. The girth is the perimeter of the end of the box. What is the maximum volume of the package with square ends that meets this criteria? ANS: 11 664 in3.
Steps:
1. Read problem carefully
2. Draw diagram showing relationship of variables
3. Define variables
4. Identify the variable to be optimized
5. Write down an equation that relates the variables listed in #3, 4
6. Differentiate (implicitly, if necessary) the equation
7. Set derivative to zero to find critical numbers
8. Use first or second derivative test to check answer optimization
9. Write answer in a complete sentence

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