Question

Construct a rectangle whose perimeter is 400cm such that it has the max area

Answer 1

Let x be the Length of the rectangle and y be the breadth of the rectangle,

Area of a rectangle A = x × y ...(1)

So, we have to maximise length and breadth of the rectangle with the constraint that 2 x + 2 y = 400

Divide by 2 on both sides,

x + y = 200

y = 200 - x...(2)

Substituting (2) in (1),

A = x (200 - x)

A = 200x - x²......(3)

Area A will be maximum at x having some value and the derivative dA/dx = 0

So differentiating (3) on both sides,

200 - 2 x = 0

x = 100

Thus, length and breadth must be 100 cm for the area to be maximum for a rectangle of perimeter 400 cm

Thus, we get a square i.e a special type of a rectangle of side 100 cm.


Here is your answer.

Answer 2

Let x be the Length of the rectangle and y be the breadth of the rectangle,

Area of a rectangle A = x × y …(1)

   

So, we have to maximize length and breadth of the rectangle with the constraint that  2 x + 2 y = 400

Divide by 2 on both sides,

x + y = 200

y = 200 - x...(2)

Substituting (2) in (1),

A = x (200 - x)

A = 200x - x²...…(3)

Area A will be maximum at x having some value and the derivative da/dx = 0

So differentiating (3) on both sides,

200 - 2 x = 0

x = 100

Thus, length and breadth must be 100 cm for the area to be maximum for a rectangle of perimeter 400 cm

Thus, we get a square i.e. a special type of a rectangle of side 100 cm