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Calculus/optimization problem

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Expert: - 10/21/2009


Question
a botanical display is to be constructed as a rectangular region with a river on one side and sidewalk 2 meters wide along the inside edges of the other 3 sides. the area for the plnats must be 800 square meters. Find the outside dimensions of the region that minimize the are of the sidewalk (and hence minimize the amount of concrete need for the sidewalk. theside walk width is x-4 and the the two outer sides inside is y-2...Thank you

Answer
One side of the building is on the river, and I'll call that L.
The width of the building I'll call W.

Since it is said that LW = 800, we know that W = 800/L.

The walkway goes at both ends with length 2L, once away from the river, with length W,
and a corner at either end.  The sidewalk has width 2, so that total area is 4L+2W+corners.
The corners have area 2x2 = 4, and there are two of them.  This says the area is A = 2L + 4W + 8.

Now to replace W with 800/L gives us A(L) = 2L + 4(800/L) + 8.

From here, we can say that A'(L) = 2 - 3200/L².

Setting this to 0 gives 2 = 3200/L².
So multiply both sides by L²/2 and get L² = 1600.
It is known that 4²=16, so 40²=1600, so L=40.

Givien that LW = 800 and L=40, we can see that W=20 since 20•40=800.

The sidewalk area is then 2L + 4W + 8 = 2•40 + 4•20 + 8 = 80 + 80 + 8 = 168.

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Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology.

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