Question

Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r.

Answer 1

Answer:

Step-by-step explanation:

Given that a rectangle is inscribed in a circle of radius r.

Since diameter subtends right angle at the circumference we get diagonal of the rectangle is diameter of the circle

Let l and w be the dimensions

Then $l^2+w^2 = 4r^2$

Hence A =lw maximum when A square is maximum

i.e. $A^2=l^2 w^2 = l^2 (4r^2-l^2)\\= 4r^2l^2 -l^4$

Use derivative test

I derivative = $8r^2l-4l^3=0\\l=0 or l = \sqrt{2} r$

II derivative <0 for non zero value of l

Hence maximum when $l=\sqrt{2} r\\w=\sqrt{2} r$

Or rectangle is a square with equal sides