# A long rectangular sheet of metal, 12cm wide, is to be made into a rain gutter by turning up two sides which make an angle of 120 degrees with the base. How many cm should be turned up to give the gutter its greatest capacity?

Nov 30, 2015

First, draw a sketch of the problem

#### Explanation:

This is an isosceles trapezoid with equal sides BC and AD $= x$

The length base CD $= {b}_{1} = 12 - 2 x$

If you draw a perpendicular from C to BA, that will be the height $h$

$h = x \sin 60$

Length of base BA $= {b}_{2} = \left(12 - 2 x\right) + 2 x \cos 60$

Area of trapezoid $= \left(\frac{1}{2}\right) \left({b}_{1} + {b}_{2}\right) \left(h\right)$

Insert the known values into the area formula ...

Area$= \left(\frac{1}{2}\right) \left(12 - 2 x + 12 - 2 x + 2 x \cos 60\right) \left(x \sin 60\right)$
Area$= \left(12 x - \frac{3}{2} {x}^{2}\right) \left(\frac{\sqrt{3}}{2}\right)$

Now, take the derivative with respect to x and set equal to zero ...

$y ' = \left(12 - 3 x\right) \left(\frac{\sqrt{3}}{2}\right) = 0$

Solution : $x = 4$ cm