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.
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.