Find the dimensions of the rectangle of area 96 whose perimeter is the least. Also, find the perimeter of the rectangle.

Given,

• Area of the rectangle is 93 cm².

• The perimeter of the rectangle is also fixed.

Let us consider,

• x and y be the lengths of the base and height of the rectangle.

• Area of the rectangle = A = x × y = 96 cm²

• Perimeter of the rectangle = P = 2 (x + y)

As,

x × y = 96

------ (1)

Consider the perimeter function,

P = 2 (x + y)

Now substituting (1) in P,

----- (2)

For finding the maximum/ minimum of given function, we can find it by differentiating it with x and then equating it to zero. This is because if the function f(x) has a maximum/minimum at a point c then f’(c) = 0.

Differentiating the equation (2) with respect to x:

[Since and ]

----- (3)

To find the critical point, we need to equate equation (3) to zero.

As the length and breadth of a rectangle cannot be negative, hence

Now to check if this critical point will determine the least perimeter, we need to check with second differential which needs to be positive.

Consider differentiating the equation (3) with x:

[Since and ]

------ (4)

Now, consider the value of

As , so the function P is minimum at .

Now substituting in equation (1):

[By rationalizing he numerator and denominator with ]

Hence, area of the rectangle with sides of a rectangle with is 96cm² and has the least perimeter.

Now the perimeter of the rectangle is

The least perimeter is .