Question

Theorem: For any given area A, the rectangle that has the least perimeter is a square.

Part I:

Complete the proof that is outlined below. Submit each step as part of your final answer.

Proof:

In terms of the side length, x, the formula for the perimeter of a square: P= 2x+((2A)/x)

Define a new variable: v=sqrt(x)-(sqrt(A))/(sqrt(x))

In terms of the new variable, v, compute and simplify: 2v^2 = _____.
Rewrite this equation to get the formula for P alone on one side. Replace the formula by the variable P. Write that result by completing the equation below:

P = 2v^2

Answer 1

Let the given area be A, this is constant
length and breadth of rectangle be given by l and b respectively
then the Perimeter is\[P=2(l+b)\]
This equation contains 2 variables, we shall eliminate one using the constant we are given
\[A=lb\]\[b=\frac{A}{l}\]Then P is a function of l is\[P(l)=2(l+\frac{A}{l})\]
Use the second derivative test, find \[\frac{dP}{dl}\] equate it to 0, find the value of l for which there is a critical point, then find \[\frac{d^2P}{dl^2}\] then evaluate the second derivative for that value of l which u calculated in first derivative, if the value is less than 0, perimeter is maximum, otherwise it is minimum

Answer 2

using the results from 2nd order test ull be able to prove what is required

Answer 3

@Nishant_Garg I love you. You are my favorite person in the entire world right now haha. Thank you so much!!!

Answer 4

Haha, thank me after you've solved the question rather! :)