Question

The cross section of a trough is an inverted isosceles triangle. Prove that the trough has maximum capacity when the vertex angle is pi/2rad.

Answer 1

Define the cross sectional area in terms of the vertex angle (assuming the sides are a constant length. Differentiate to find the stationary points and prove pi/2 is a max.

Answer 2

Note that the area of a triangle with sides a,b,c, with the angle C between a and b has an area:
\[\text{Area} = \frac{1}{2}\cdot a\ \cdot b\ \cdot \sin C \]

Answer 3

so differentiate 1/2ab sinc? that become 1/2ab cosc??