4.5.8a-eg-1 - Example A graphics designer plans to close...

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Unformatted text preview: Example A graphics designer plans to close off a corner of the first quadrant of the x- y- plane with a line segment that is 16 cm long, by connecting a point ( x, 0) on the x-axis to a point (0 ,y ) on the y-axis. How should x and y be selected in order to maximize the area inside the triangle? Solution: Since x is the length of the base of the right triangle, and y is its height, the goal is to maximize the area of the triangle given by Area = 1 2 xy. In order to apply the methods of first semester Calculus, we need to rewrite this expression as a function of one variable only. Now, the length of the hypotenuse of the triangle must always be 16 cm, so this is a constraint that relates x and y and will allow us to solve for y in terms of x . That is, x 2 + y 2 = 16 2 , or y = 256- x 2 . (1) Thus the goal is to minimize the function A ( x ) = 1 2 x 256- x 2 , where x is constrained to satisfy x [0 , 16]. (Note that x = 0 or x = 16 are not physically reasonable choices, however they are not values which are likely to maximize the area in any...
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This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.

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4.5.8a-eg-1 - Example A graphics designer plans to close...

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