Question

Find the rectangular shape of area A= 25 and minimum perimeter

Answer 1

Let x and y be the length and breadth of the rectangular shape respectively.

Then Area is A=xy ...........(1)

also given area A=25 ..........(2)

from (1) and (2) we get xy=25

form here we find y=25/x

Perimeter is given by P=2(x+y)=2x+2y

use y=25/x

we get P=2x+2×(25/x)=2x+50/x

Now diffrentiate w.r.t to x we get

P'=2-50/x^2

put P'=0 we get 2-50/x^2=0

2=50/x^2

x^2=25

so we get x=5

so from y=25/x=25/5=5

we check perimeter is minimum by double derivative test

P"=100/x^3 >0 at x=5 ,hence minimum

Hence rectangular shape (whose lengh is x=5 and breath is y=5) is Square.

Answer =Square of length of each side 5