Maximizing Window AreaDate: 02/24/97 at 20:37:16 From: pierre Subject: Related rates The business office wants to put in Norman windows, which are windows in the shape of rectangles capped by a semicircle. These windows will have special trim that is purchased by length, so it is most cost efficient to get the most light for the least perimeter of window. What dimensions maximize the area of the windows? Date: 02/25/97 at 19:32:55 From: Doctor Luis Subject: Re: Related rates Pierre, I'll try to draw the window, just to visualize the problem: * * * * (circle w/ radius r) *_____________* | | | | height h | | |_____________| length 2r Obviously, the larger the dimensions of the window are, the larger area you'll get. So the maximum area possible is obtained when both r and h (refer to diagram) are as large as possible. Of course, that depends on how much money you have to spend, which fixes a value for the perimeter of the window. The problem, then, becomes: "For a given perimeter p find the dimensions h and r which will give the greatest area for our window" An expression for the area of the window is: a = area(window) = area(rectangle) + area(semi-circle) = 2rh + (pi*r^2)/2 [remember: r^2 means "r squared"] An expression for the perimeter p of the window is: p = p(window) = p(rectangle) + p(semi-circle) - 2*(overlapping side) = (4r+2h) + (pi*r+2r) - 2*(2r) = 2h + (pi+2)r Since a = a(r,h) is a function of two variables, we must look for a relationship between r and h in order to make the problem easier. Since p is a constant (it's given), r and h can't just have any values. They must satisfy the expression for p derived above. Now, solving for h, we have: h = (p-(pi+2)r)/2 Substituting this expression for h into a, we obtain: a = 2r(p-(pi+2)r)/2 + (pi*r^2)/2 a = r(p-(pi+2)r) + (pi*r^2)/2 Now, a is a function of a single variable (a = a(r)), so we can apply the methods of maxima and minima learned in calculus to find out which value of r will maximize a(r): i) set a'(r) = 0 and solve for r to find critical points ii) figure out the sign of a''(r) at the critical points to find out if they are maxima or minima (or points of inflection). I'm sure you can work the rest of the problem now. :) (By the way, the critical point I found is r = p/(4 + pi).) -Doctor Luis, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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