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Calculus/differentiation

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Expert: - 9/3/2010


Question
An architect plans to build a triangular enclosure with a fence on two sides and a wall on the
third side.Each of the fence segments has fixed length L.What should be the length x of the
wall side,expressed in terms of L,so that the region enclosed has the largest possible area?

Answer
Let A be the angle.  Cut A in half and form a right triangle with the fence of length L and the half of the far side, S/2.

The distance from where the two fences meet to the middles of the side of unknown length is [(L/2)^2 - (S/2)^2]^(1/2).

The total area would be the sum of the areas of the two identical triangles.  That is (S/2) [(L/2)^2 - (S/2)^2]^(1/2).

Note that S is the variable and L is a constant.

To solve, take the derivative and set it to 0, the find S in terms of L.

Note that the derivative is a product rule between F(S) = S/2 and
G(S) = [(L/2)^2 - (S/2)^2]^(1/2).

Note that the derivatives of each of these is F'(S) = 1/2 and
G'(S) = ((S/2)/[(L/2)^2 - (S/2)^2]^(1/2)).

The product rule says the derivative of the whole function is
F(S)G'(S) + F'(S)G(S).

Calculus

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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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