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ep10 - 92.131 Calculus 1 Optimization Problems 1 A Norman...

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92.131 Calculus 1 Optimization Problems 1) A Norman window has the outline of a semicircle on top of a rectangle as shown in the figure. Suppose there is π + 8 feet of wood trim available for all 4 sides of the rectangle and the semicircle. Find the dimensions of the rectangle (and hence the semicircle) that will maximize the area of the window. 2) You are building a cylindrical barrel in which to put Dr. Brent so you can float him over Niagara Falls. I can fit in a barrel with volume equal 1 cubic meter. The material for the lateral surface costs $18 per square meter. The material for the circular ends costs $9 per square meter. What are the exact radius and height of the barrel so that cost is minimized? 3) A rectangular sheet of paper with perimeter 36 cm is to be rolled into a cylinder. What are the dimensions of the sheet that give the greatest volume? 4) A right triangle whose hypotenuse is 3 m long is revolved about one of its legs to generate a right circular cone. Find the radius, height, and volume of the cone of greatest volume. Note: h r V 2 3 1 π = . 5) Determine the cylinder with the largest volume that can be inscribed in a cone of height 8 cm and base radius 4 cm . 3 h r
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