Q. 27

A rectangle is in

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.

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