# Thread: optimization problem

1. ## optimization problem

Determine the area of the largerst rectangle that can be inscribed in a right cemicircle with a radius of 10 units. Place the length of the rectangle along the diameter.
Thank you in advance!

2. Originally Posted by anna12345
Determine the area of the largerst rectangle that can be inscribed in a right cemicircle with a radius of 10 units. Place the length of the rectangle along the diameter.
Thank you in advance!

Start with a picture, with the semi-circle centered at the origin. Let top right corner of the rectangle have coordinates (x, y) where both x and y are positive (first quadrant).

$\displaystyle x^2 + y^2 = 100$ since the point is located on the circle

The area is $\displaystyle A = (2x) * y = 2xy$

Substitute into the area function so it depends on x or y, only. Then differentiate the area function to find either $\displaystyle dA/dx$ or $\displaystyle dA/dy$. Then solve for critical numbers. You may want to verify that your solution is a max (and not a min) using one of the derivative tests.

Hope this helps!