Given,

• Radius of the semicircle is ‘r’.

• Area of the rectangle is maximum.

Let us consider,

• The base of the rectangle be ‘x’ and the height be ‘y’.

Consider the ΔCEB,

CE² = EB² + BC²

As CE = r, and CB = y

---- (1)

Now the area of the rectangle is

A = x × y

Squaring on both sides

A² = x² y²

Substituting (1) in the above Area equation

----- (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 Z(x) has a maximum/minimum at a point c then Z’(c) = 0.

Differentiating the equation (2) with respect to x:

[Since ]

------- (3)

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

x(2r² – x²) = 0

x = 0 (or) x² = 2r²

x = 0 (or)

[as x cannot be zero]

Now to check if this critical point will determine the maximum area, we need to check with second differential which needs to be negative.

Consider differentiating the equation (3) with x:

----- (4)

[Since ]

Now let us find the value of

As , so the function Z is maximum at

Substituting x in equation (1)

As the area of the rectangle is maximum, and and

So area of the rectangle is

A = r²

Hence the maximum area of the rectangle inscribed inside a semicircle is r² square units.