Sample-free-response-final-solutions

Sample-free-response-final-solutions - Math 1314 Sample...

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Math 1314 Sample Free Response Questions Final Exam Optimization problems 1. To ship a package through the postal service, the length plus girth of the package can be no more that 108 inches. You want to design a package that is a box with a square base so that the volume of the box is maximized. Find the dimensions of the box. Note: girth is the perimeter of the base of the package. Draw a picture: y x x We want to maximize the volume of this box. We’ll use the volume formula, lwh V = . For our box, y x V 2 = . We also know that the length plus girth cannot exceed 108 inches. That is y x 4 can be no more than 108. We want to maximize volume, so we’ll want to use all of that 108 inches. So we have 108 4 = y x . Solve this for y and substitute it into the volume formula: 3 2 2 4 108 ) 4 108 ( ) ( 4 108 108 4 x x x x x V x y y x - = - = - = = Domain: 0 x and 0 4 108 - x . Solve the second inequality: 27 x . We want to find the absolute maximum of 3 2 4 108 ) ( x x x V - = on [0, 27]. Find the derivative: 2 3 2 12 216 ) ( ' 4 108 ) ( x x x V x x x V - = - = Find any critical numbers:
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18 , 0 0 ) 18 ( 12 0 12 216 2 = = = - = - x x x x x x Substitute the critical numbers that are in the interval and the endpoints of the interval into the function V ( x ) and evaluate. x V(x) 0 0 18 11664 27 0 So volume is maximized when 18 = x . To find y , substitute 18 into the statement x y 4 108 - = and evaluate. So y = 36. The dimensions of the box are 18 inches by 18 inches by 36 inches.
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2. Suppose you want to enclose a rectangular region that is located along the straight edge of a river. The side along the river does not need to be fenced. What is the largest possible area that you can fence is if you have 1200 meters of fencing to use? What would be the dimensions of the region. Start with a sketch: y x x River We want to maximize the area of this region. Area is lw A = and in our case, that will be xy A = .
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