Question

A rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle so that its area is maximum. Find also this area.

Answer 1

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

Now the area of the rectangle is

A = x × y

Squaring on both sides

A² = x² y²

Substituting (1) in the above Area equation

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:

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

[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:

Now let us find the value of

so the function Z is maximum at x = r√2

Substituting x in equation (1)

As the area of the rectangle is maximum, and x = r√2 and y = r√2/2

So area of the rectangle is

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