Find the dimensions of the rectangle with perimeter 200 mete

Solution

let ;
x=heigth
y=width

Perimeter will be

2x+2y=200 => x+y=100----(1) y=100-x
Area of rectangle will be xy
from equation 1 we can write that x(100-x)

we can write it as a function of A(x)

A(x)=x(100-x) = 100x-x^2
now will try to find the highest x in this constraint which will be optimized solution. So finding critical point and proving it as maximum will do our job. So we are gonna use Calculus:-

taking derivative
A\'(x)=100-2x
A\'(x)=0=100-2x
2x=100 => x=50
taking another derivative
A\'\'(x)=-2 <0 so concave down all the time so x=50 will be maximum

y=100-50= 50

so the dimension will be x=50 y=50