Question

What is the maximum area of the rectangle with perimeter 500 mm?

(a) 15,625 mm^2

(b) 15,025 mm^2

(c) 15,600 mm^2

(d) 10,625 mm^2

This question was addressed to me in an online quiz.

My question comes from Total Derivative topic in portion Partial Differentiation of Engineering Mathematics

Answer 1

Correct answer is (a) 15,625 mm^2

Best explanation: 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)

500=2(x+y)

x+y=250

y=250-x

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

A=x*(250-x)

A=250x-x^2

To find maximum value we need derivative of A,

\(\frac{dA}{dx}=250-2x\)

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

250-2x=0

2x=250

x=125 mm

Therefore, when the value of x=125 mm and the value of y=250-125=125 mm, the area of the rectangle is maximum, i.e., A=125*125=15,625 mm^2