Question

Find the dimensions of the rectangle of maximum area that can be inscribed in a circle of r=61

Answer 1

from trig (unit circle) you know that
x = rcos(angle)
y = rsin(angle)

\[Area = 4r^{2} \cos \theta \sin \theta\]
to maximize Area , set derivative equal to 0
\[\frac{dA}{d \theta} = 4r^{2}(\cos^{2} \theta - \sin^{2} \theta) = 0\]
solve
\[\theta = \frac{\pi}{4}\]
\[Area_{\max} = 4r^{2}\cos (\frac{\pi}{4}) \sin (\frac{\pi}{4})\]

Answer 2

Okay I understand that now! The question is now asking to "(Use symbolic notation and fractions where needed. Give your answer in the form of comma separated list of the dimensions of the two sides.)" I'm confused though, wouldn't the sides be 61 and 122?

Answer 3

well the dimensions are same since sin(pi/4) = cos(pi/4)
\[2r*\frac{\sqrt{2}}{2} = \sqrt{2}r\]