This math problem popped up in my textbook:
A strip of metal is 24cm wide. This strip of metal is to be used to
form a length of guttering for a house. The guttering will be open at
the top so water can flow through. What are the dimensions of the
cross section of the guttering if it is to hold the maximum volume of
water?
I solved it, and got the correct answer of Width = 12cm, Depth = 6cm (therefore you will have to fold up 6cm of metal on each side).
However, what I cant understand is why a square wouldn't give maximum volume. It is often considered to be a given that to maximise area, a rectangle of equal length sides (i.e. A square) should be used. However, why is it that in this problem a rectangle is the optimum shape?
Best Answer
Intuitively, it is because you don't need material to cover the top of the gutter, so width costs half as much as height. This should cause the solution to be wider than a square. A clever approach is to note that if you make a copy of the gutter, invert it, and put it on top of your gutter, you have the classic problem of maximizing the area of a rectangle for a given perimeter. As you say, this is solved by a square, so the gutter problem should be twice as wide as it is high. All this is a bit of handwaving, and mvw's answer is spot on.