Using the derivative of trigonometric function.
A trapezoidal gutter is to be made from a strip of metal 22 inches wide, by bending up the edges. If the base is 14 inches wide, what is the width accross the top gives the greatest carrying capacity.
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You should draw an isosceles trapezoid, whose base is of 14 inches, hence, the lateral sides measure each 4 inches.
You should evaluate the area of trapezoid such that:
`A = (1/2)h(B+b)`
You should come up with the following notation of the angle made by the height of trapezoid to lateral side: `alpha` .
You may evaluate the height using the rigth triangle whose hypotenuse is the lateral side of 4 inches, such that:
`cosalpha= h/4 => h = 4 cos alpha`
`B = 14 + 2*4 sin alpha`
`A = (1/2)*4cos alpha(14 + 14 +8 sin alpha)`
`A = (1/2)*4cos alpha(28 + 8 sin alpha)`
`A = 56cos alpha + 8sin 2alpha`
You need to optimize the area, hence, you should differentiate the function A with respect to `alpha` such that:
`A' = -56sin alpha + 16cos 2alpha`
Solving the equation A' = 0 yields:
-56sin alpha + 16cos 2alpha = 0
`-14sin alpha + 4cos 2alpha = 0`
`-7sin alpha + 2cos 2alpha = 0`
Substituting `1 - 2 sin^2 alpha` for `cos 2alpha` yields:
`-7sin alpha + 2 - 4sin^2 alpha = 0`
`4sin^2 alpha + 7sin alpha - 2 = 0`
Using quadratic formula yields:
`sin alpha_(1,2) = (-7+-sqrt(49 + 32))/8`
`sin alpha_(1,2) = (-7+-sqrt81)/8`
`sin alpha_(1,2) = (-7+-9)/8 => sin alpha_1 = 1/4; sin alpha_2=-2`
`sin alpha_2 = -2 ` is invalid
`sin alpha_1 = 1/4 => alpha = sin^(-1)(1/4) ~~ 14^o`
You may evaluate the width accross the top such that:
`B = 14 + 8 sin alpha => B = 14 + 8*(1/4) => B = 16` inches
Hence, evaluating the length of the width that gives the greatest capacity, under the given conditions, yields B = 16 inches.
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