• Oct 26th 2007, 08:31 AM
runner07
how would you do this....

Find the largest area of a rectangle inside a circle with a radius of 4.
• Oct 26th 2007, 10:01 AM
ticbol
It should be a square. :)

Draw the figure on paper, as I cannot show it here.
Let x = width of the rectangle
And y = the length
Draw any diagonal. This diagonal is a diameter of the circle so it is 2r long.

Area of rectangle, A = xy

By Pythagorean theorem, x^2 +y^2 = (2r)^2
x^2 +y^2 = 4r^2
y^2 = 4r^2 -x^2
y = sqrt(4r^2 -x^2)

So,
A = x*sqrt(4r^2 -x^2)
Differentiate both sides with respect to x, (r is a constant),
dA/dx = x[-2x / 2sqrt(4r^2 -x^2)] +sqrt(4r^2 -x^2)
Set that to zero,
0 = x(-x) +[sqrt(4r^2 -x^2)]^2
0 = -x^2 +4r^2 -x^2
2x^2 = 4r^2
x^2 = 2r^2
x = r*sqrt(2) ------***
So,
y = sqrt[4r^2 -2r^2] = sqrt[2r^2] = r*sqrt(2) also.

Therefore, the largest area of a rectangle in a circle whose radius is 4 is
A = 2sqrt(2) *2sqrt(2) = 4*2 = 8 sq.units ------------answer.
• Oct 27th 2007, 05:05 AM
qyzren
Area is not 8 sq units...

However ticbol is correct about it being a square.
Here's a simple way of doing the question.
http://img222.imageshack.us/my.php?i...ture161yg3.jpg