Question

What is the maximum area of the rectangle with perimeter 650 mm?

(a) 26,406.25 mm^2

(b) 26,406 mm^2

(c) 24,000 mm^2

(d) 24,075 mm^2

I got this question during a job interview.

This key question is from Geometrical Applications topic in section Ordinary Differential Equations – First Order & First Degree of Engineering Mathematics

Answer 1

Correct choice is (a) 26,406.25 mm^2

The explanation is: Let x be the length of the rectangle and y be the width of the rectangle. Then, Area A is,

A=x*y              …………………………………………………. (1)

Given: Perimeter of the rectangle is 620 mm. Therefore,

P=2(x+y)

650=2(x+y)

x+y=325

y=325-x

We can now substitute the value of y in (1)

A=x*(325-x)

A=325x-x^2

To find maximum value we need derivative of A,

\(\frac{dA}{dx}=325-2x\)

To find maximum value, \(\frac{dA}{dx}=0\)

325-2x=0

2x=325

x=162.5 mm

Therefore, when the value of x=162.5 mm and the value of y=325-162.5=162.5 mm, the area of the rectangle is maximum, i.e., A=162.5*162.5=26,406.25 mm^2