Let x be the width of the base, y the height. Then the length of the base is 2x and the volume= so .

The area of the base is so the cost of the base is dollars.Material for the base costs $10 per square meter.

Two of the sides have area . The cost of those two sides is .Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container.

Two of the sides have area . The cost of those two sides is .

The total cost of the container is .

That is the function you want to maximize.

The gutter will "carry the maximum amount of water" when the area of the triangle formed is maximum. If you have an isosceles triangle with sides of length 30/2= 15 cm and vertex angle , then you have two right triangles with hypotenuse of length 15 and angle . The "opposite side", which is at the top of the gutter, is so the width of the entire gutter, the "base" of the triangle, is . The "near side", the altitude of the entire triangle, is .I really don't see what equations I could get out of that information.

While I am at it, I have never worked with optimization problems that involve radians.

A rain gutter is to be constructed from a metal sheet of width 30 cm by bending up one-third of the sheet on each side through an angle θ. How should θ be chosen so that the gutter will carry the maximum amount of water?

Again I just need someone to point me in the right direction. Namely, how and what the equations should be.

Since "Area= (1/2)base*height" for a triangle, the area of the triangle formed by this gutter, and the function you want to maximize, is .