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A rain gutter is to be made of aluminum sheets that are 12 inches wide by turning up the edges 90 degrees What depth will provide maximum cross-sectional area and hence allow the most water to flow?
Wiki User
∙ 14y ago
You really should consider trying this yourself before reading the answer, and you should understand the answer, otherwise you will not learn the lesson. Read each step, and do not proceed unless you understand it.
To maximize the cross sectional area of a rain gutter made from sheets that are 12 inches wide by turning up the edges 90 degress, first draw the cross section...
The depth is x. The cross section is 12-2x by x. Try again to draw it...
The area of the cross section is (12-2x)x. Can you see the solution?
Simplify the area. It is 12x - 2x2. Can you see the solution?
Differentiate. This is a simple polynomial. Can you see the solution?
The deriviative is dy/dx = 12 - 4x. Can you see the solution?
The maximum volume occurs when the deriviative is zero. Can you see the solution?
Solve for 12 - 4x = 0. Last chance to solve it yourself...
There is one root of x: 3, so the depth is 3 inches.
Wiki User
∙ 14y ago
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