8. A rectangular sheet of metal 20 feet long and 2 feet wide is bent down the middle to form a "V". Two triangular pieces of metal are soldered onto the ends to form a prismatic trough as in the picutre to the right. What does the height of the triangle have to be to maximize the volume of the trough?
Since the figure is a prism, the formula for the volume of a prism is
where s is the length of the trough, and A is the area of the triangular end piece. using the formula for the area of a triangle we get
If you look at the triangluar base
we see first, that since the piece of metal is 2 feet wide, and we are folding it down the middle, the two sides of the triangle will both have a length of 1 foot. The width of the triangle at the top is the base in the formula for the area. We have an isosceles triangle, and the height divides it into two congruent right triangles.
We can use the Pythagorean Theorem to find half the base.
As a result we can express the volume as a function of the desired unknown h.
To maximize the volume, we differentiate ith respect to h and set the derivative = 0.
Since we have a product, we use the product rule to differentiate.
This simplifies to
Factor out the 20 and find common denominators.
A fraction is 0 when its top is 0
Which is well known to be half the diagonal of a square. The cross section of the trough of maximum volume will look like
where the angle at the bottom of the trough is a right angle.
This makes sense. If you took another identical trough and put it on top of the first one to build a rhomboidal prism, you would maximize the double the volume when you maximized the volume of the original prism. The volume of the rhomboidao prism is maximized then the cross sectional rhombus has maximum area. It is well known that the rhombus of given perimeter with the maximum area is the square.