UFCalcSet28 - Exercises UF Calculus Set 28 1. Solve the...

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Exercises UF Calculus Set 28 1. Solve the following arithmetic optimizations: (a) Split 20 into two nonnegative numbers x,y such that the product of x and y 2 is a maxi- mum. (b) Find two nonnegative numbers x,y whose sum is 8 and for which x 2 + y 2 is a minimum. (c) Find two numbers x,y 1 whose product is 50 and for which 2 x + y is a maximum. 2. A rectangular box which is open at the top can be made from a 18 by 12 inch piece of metal by cutting a square from each corner and bending up the sides. Find the dimensions of the box with greatest volume. 3. A poster of total area 400 in 2 is to have a margin of 4 inches at the top and bottom and 1 inch at each side. (a) Find the dimensions which give the largest printed area. (b) Suppose that, instead, we know the poster will contain 400 in 2 of printed material with margins of 4 inches at the top and bottom and 1 inch at each side. Find the dimensions of the poster that minimize its total size. 4. Suppose you want to reach a point A that is located across the sand from a nearby road. Suppose that the road is straight, and 2 miles is the distance from A to the closest point C on the road. Suppose 60 mph is your speed on the road, and 30 mph is your speed on the sand. Right now you are at the point D , which is a distance 5 miles from C . At what point B should you turn off the road and head across the sand in order to minimize your travel time to A ? What should happen if your speed on the sand is at least as great as your speed on the road? 5. Find the dimensions of the right triangle of maximum area whose hypotenuse has length one. 6. A wire of length L is to be divided into two parts; one part will be bent into a square and the other into a circle. How should the wire be divided to make the sum of the areas of the square and circle as large as possible? as small as possible? 7. Solve the problems below that pertain to inscribing one figure in another. (a) Find the area of the largest rectangle which can be inscribed in a semicircle of radius r (b) Find the fraction of the area of a triangle that is occupied by the largest rectangle that can be drawn in the triangle (with one of its sides along a side of the triangle). Show that
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This note was uploaded on 04/04/2012 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.

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UFCalcSet28 - Exercises UF Calculus Set 28 1. Solve the...

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