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what are the dimensions of a rectangle with the largest area that can be drawn inside a circle of radius 2

User Turismo
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1 Answer

7 votes

Answer:

2r2units

Explanation:

Let a rectangle of dimensions x×y

Be inscribed in a circle of radius r.

Applying Pythagoras theorem on the right-angled triangle,

⇒x2+y2=(2r)2⇒x2+y2=4r2⇒y2=4r2−x2⇒y=4r2−x2−−−−−−−√

As dimensions of any shape cannot be negative so y=−4r2−x2−−−−−−−√

is not considered.

Area of the rectangle,

⇒A=x×y⇒A=x4r2−x2−−−−−−−√

Differentiating both sides with respect to x, we get –

dAdx=ddx(x4r2−x2−−−−−−−√)

Use the product rule to differentiate the above result –

A1(x)=xd(4r2−x2−−−−−−−√)dx+4r2−x2−−−−−−−√d(x)dxA1(x)=xd4r2−x2−−−−−−−√d(4r2−x2)d(4r2−x2)dx+4r2−x2−−−−−−−√(1)A1(x)=x124r2−x2−−−−−−−√(−2x)+4r2−x2−−−−−−−√A1(x)=−x2+4r2−x24r2−x2−−−−−−−√A1(x)=−2x2+4r24r2−x2−−−−−−−√

Now to get the maximum value, put A1(x)=0

−2x2+4r24r2−x2−−−−−−−√=0⇒4r2=2x2x2=2r2x=2–√r

The negative value of x is rejected.

⇒y=4r2−x2−−−−−−−√=r2–√

A(r2–√)=r2–√(4r2−2r2−−−−−−−√A(r2–√)=r2–√(r2–√)=2r2

The maximum area of a rectangle inscribed in a circle is 2r2units

and its dimensions are r2–√units×r2–√units

So, the correct answer is “ 2r2units ”.

User Callahad
by
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