Find the area of the largest rectangle that can be inscribed in a semicircle of radius 5.

The area of the largest rectangle is 25 square units.

Step-by-step explanation:

It is given that the radius of the semicircle is .

It is required to inscribe a rectangle of maximum area.

Consider a rectangle of sides and as shown in the given figure.

From the attached figure, , and .

Now, in the triangle OAB, apply Pythagoras theorem as,

So, the area of the rectangle will be,

Differentiate the area with respect to ,

Differentiate the area twice as,

So, the double derivative of area function is negative and hence, the area will be maximum at .

Therefore, the maximum area of rectangle will be,

Therefore, the area of the largest rectangle is 25 square units.

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Maximum area of rectangle = 50 sq unit

Step-by-step explanation:

A rectangle inscribed in a semi-circle of radius 5.

Please find attachment for figure.

Let length of rectangle be 2y and width be x .

In ΔOAB, ∠OBA = 90°

    (Using pythagoreous theorem)

Area of rectangle = Length x Width

                         A = 2y × x

Derivative above function w.r.t x

For maxima/minima

To check maximum/minima using double derivative test

At  ,

Hence, the area is maximum when width, and

Maximum area of rectangle, A= 2xy

step-by-step explanation:

b

step-by-step explanation: