Question

"Explain the statement:
Of all rectangles with a fixed perimeter, the square measures out to the largest area."

How would you go about proving this to be true?

Answer 1

I guess I cant use the diagonal.. they're not consistent across different base & height.

Answer 2

GIVEN
area = base * height
perimeter = 2base + 2height
EQUATIONS
area[x,y] = xy
perimeter[x,y] = 2x+2y

Answer 3

So maybe I need to prove that the argument of the statement is that base/height=1 will yield the highest value for base*height

Answer 4

Let define the fixed perimeter , call it C
C = 2x + 2y
We want to maximize the area of the rectangle.
Area (A) of a rectangle is :

A = x * y
Area is now a function of two variables x and y.
It would be easier to maximize the area if we could eliminate one of the variables.
How about eliminating y. Use the first equation to solve for y in terms of x.
C - 2x = 2y
(C - 2x ) / 2 = y

Now substitute.
A = x * ( C - 2x)/2

Answer 5

With me up to here?

Answer 6

I think so.. was just thinking wow...

Answer 7

Im not sure where to take it form here though..

Answer 8

At this point you can you use calculus or use your knowledge of parabolas

Answer 9

so Im guessing then a derivative ties into this somehow ..

Answer 10

wow again..

Answer 11

do we need to do the same thing for y?

Answer 12

We have a quadratic function
A = x * ( C - 2x)/2 bring the 1/2 in front
A = x/2 * (C - 2x) distribute
A = Cx/2 - 2x^2/2
A = Cx/2 - x^2
A = -x^2 + Cx/2
dA/dx = -2x + C/2

According to Fermat's theorem the extrema occurs when dA/dx = 0

dA /dx = 0
-2x + C/2 = 0

-2x = -C/2
2x = C/2
x = C/4

Plug this into C = 2x + 2y, and solve for y

C = 2(C/4) + 2y
C = C/2 + 2y
C - C/2 = 2y
C/2 = 2y
1/2 * C/2 = y
C/4 = y

So far we have x = C/4 , and y = C/4
We must have a square

Answer 13

wow.. thats amazing, I dont think I was going to work this out on my own.. thank you.

Answer 14

Here is a problem that is similar. Show that of all isosceles triangles of fixed perimeter, the triangle that has maximum area is an equilateral triangle.

Answer 15

Perimeter = x + 2y
Area = 1/2 * x * height

You can get the height using pythagorean theorem

Answer 16

once you have a solution for x that minimizes the area, you can back substitute to find the y value

Answer 17

oh cool, yeah that makes sense.

Answer 18

We assumed that x was the length and y was the width (or vice a versa) in our drawing.
Our work shows that the maximize area has length C/4 and width C/4
The only rectangle that has equal length and width is a square.

Answer 19

where C was the fixed perimeter

Answer 20

o to summarize ..
1. We redefined the perimeter as a constant C
2. Then defined area as a function of x and y, area = xy
. .. the plan here is to find the derivative and look for the maxima by Fermat's Theorem.
3. Before looking for the maxima of area, we simplify the area function to make it a function of x with the constant C, I'm guessing so we dont have to deal with a partial derivative maybe, and we have area[x]= ....
4. We distribute and simplify our area function
5. Then we work out the derivative and find area'[x]
6. Then we Solve for area'[x] ==0 and apply Fermats Theorem.
6a ... perhaps we might add here a test for 2nd derivative <0 to prove we have a maxima?
7. Then plug y back into C = 2x + 2y and solve for y
8. Finally we get two values that are equal for x and y
9. we might be able to say that as a square has sides that are b/h = 1 then because x/y=1 we have a square.

Answer 21

ah, yes, only square rect's have x=y

Answer 22

right by definition

Answer 23

well I'm good to go .. thank you.