Hello, ihmth!

1) A trapezoid gutter is to be made from a strip of metal 22 in. wide, by bending up the edges.

If the base is 14 in. wide, what width across the top gives the greatest carrying capacity?

(use trig function) Code:

F E
A * - - * - - - - - - - * - - * D
\ : : /
\ : : /
4 \ : h: / 4
\ : :θ/
\: :/
*---------------*
B 14 C

The isosceles trapezoid is $\displaystyle ABCD$, where $\displaystyle BC = 14$, and $\displaystyle AB = CD = 4.$

Let $\displaystyle \theta = \angle ECD$

Area Formula: .$\displaystyle A \;=\;\frac{h}{2}(b_1 + b_2)$

In right triangle $\displaystyle DEC\!:\;\;DE = 4\sin\theta,\;EC = 4\cos\theta$

So we have: .$\displaystyle h \:=\:4\cos\theta,\;b_1 \:=\: 14,\;b_2 \:=\:14 + 8\sin\theta$

Then: .$\displaystyle A \;=\;\frac{4\cos\theta}{2}\bigg[14 + (14 + 8\sin\theta)\bigg] \quad\Rightarrow\quad A \;=\;8\cos\theta(7 + 2\sin\theta)$

. . and __that__ is the function to be maximized . . .