Find the largest possible rectangle that can be inscribed in a circle of radius 25.

Please show all your work.

We're a place where learners ask for help for their tasks and share their knowledge.

Find the largest possible rectangle that can be inscribed in a circle of radius 25.

Please show all your work.

For further actions, you may consider blocking this person and/or reporting abuse

Catawba -

Phyllis Cook -

Phyllis Cook -

gjdsla;gjkdsa;jg -

Todd -

Joseph Farmer -

Love.Live.Laugh -

Jack T -

Mike -

Chem Help -

AmericansAreEvil -

dan -

Katherine Gonzalez -

Meowzy Marshmallow -

Christina Bailey MD -

Ozz B -

school1234 -

Evan -

Kelsey -

Dominic S -

## Answers (2)

Let's consider only one quadrant, since the total rectangle is four times that. I am also assuming you mean largest in area.

The equation of a circle is x^2+y^2 = r^2. You know that the radius is 25, so all points on the circle satisfy the equation x^2 + y^2 = 25^2.

If you pick an arbitrary point on the circle and make that one corner of your rectangle, you can multiply x and y to get the area. So if you solve the circle equation for y in terms of x, you can make an expression of volume in terms of x.

y = sqrt (25^2 - x^2) = (25^2 - x^2) ^ 1/2

This value has a maximum somewhere between x = 0 and 25. We know this because at x=0 the rectangle has no width, and at x=25, the rectangle has no height. So take the derivative of the above equation and solve for zero.

dy/dx = 1/2 (25^2 - x^2) ^ -1/2 * -2x = - x /sqrt (25^2 - x^2); this value of x is half the width of your rectangle and I am guessing it will be half sqrt(2) * 25. Don't forget to multiply by four.

A square with side length 50/sqrt(2) is the largest area rectangle that will fit.