Answer to September 2, 2002 Problem |
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Don't Fence Me In! |
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What is the largest possible total area of the four pens?
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Solution to Problem: The answer is 14,062.5 square feet. You can solve the problem with calculus or with precalculus. I will outline the precalculus solution:
Let w = the total width of the four pens.
Solve the first equation for w:
Now substitute this expression for w in the Area equation:
Simplifying,
Perhaps, it is easier to see this if you replace A by y. y = 375x - 2.5 x^2 You can solve this on a graphing calculator. If you do, be sure to change the domain and range!!! (Use something like 50 to 100 for x; and use 12000 to 15000 for y). You can also solve this by completing the square. This will give you the coordinates of the vertex (x, y), which will give you the length of each pen (x) and the area of the four pens (y), which, of course, is what you are looking for. y = 375x - 2.5x^2
I would rewrite it as: 2.5 (x^2 - 150x ) = -y 2.5 (x^2 - 150x + 5625) = -y + 2.5(5625) 2.5 (x^2 - 150x + 5625) = -y + 14,062.5 2.5 (x - 75)^2 = - (y - 14,062.5) -2.5 (x - 75)^2 = y - 14,062.5
so the vertex is (75, 14062.5) |
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