Posted by Muffy on Sunday, November 1, 2009 at 1:01pm.
Good question!
2X+2Y could be equal to lots of things: your question is to find the _minimum_ they can be equal to.
For example, we know
xy=400
so x=5, y = 80 works for that.
Then 2x + 2y = 10 + 160 = 170. That is one answer.
x=10, y=40 also gives us xy=400.
Then 2x + 2y = 20 + 80 = 100. That is another answer, and a better one than 170, since it's smaller, and therefore closer to the minimum.
A lot of calculus is about finding the minimum or maximum value, which is often the "best" or "worst" case in real life, but we can't use calc here, so can you find another set of values for x and y so that
x * y = 400
and 2x + 2y is as small as possible?
Let me phrase it as a real-life-ish word puzzle:
You need to build a pool 400 sq. m. in area. Building curved or slanted sides is too complicated and expensive, so the sides have to be straight, so you're going to make it a rectangle.
The sides cost you $10 per metre to put up.
What are the lengths of the sides of the cheapest pool you can build?
So basically, I have to guess and check.
20 X 20 would be the smallest but is a square.
How about 16 X 25 = 82?
I would much rather have a mathematical way to go about it.
First thing to remember is that a square is a rectangle; it's just a special case of a rectangle.
Getting back to a "mathematical way to go about it", there are two I can think of, and one involves calculus. Maybe this is set to make you appreciate calculus when you get to it. :-)
Anyway consider that since we know:
xy = 400
we know that
y = 400 / x
which means that we can restate
2x + 2y
as
2x + 2(400 / x).
So we can restate the problme as finding the minimum value of
x + 400 / x
Does that prompt any new ideas?
Ok, well now I'm really confused. I thought about substituting 400/x into y but I still don't have anything to put it equal to.
Can you please try to explain it.
Actually, putting 400/x into y will give you something, here's a geometrical approach of finding the minimum.
Let x stands for one side of the "rectangle", not forgetting that a square is also a rectangle, as Jim has pointed out. The adjacent side will be 400/x, thus assuring that the product of the two sides will give an area of 400.
The perimeter, P is therefore given by
P = 2(x+400/x)
For x to be the value for which the perimeter is the minimum, then any perturbation, h, in the value of x will result in a bigger perimeter, i.e.
2((x+h)+400/(x+h))≥2(x+400/x)
for any value of h, i.e. h≥0 or h≤0.
Expanding and simplifying, we obtain
h(1-400/(x(x+h)))≥0
if h≥0, x(x+h)≥400
if h≤0, x(x+h)≤400
The only value of h that satisfies both inequalities is h=0, or x²=400, hence h=20.
If you would like to have some practice of the method, you can try to find the minimum perimeter of three sides of a rectangle of a given area, A. The problem arises when we have to fence an area against a wall. You will find that the two opposites sides are √(A/2), and the side opposite the wall is 2√(A/2).