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!