SOLUTION: Find the dimensions of the rectangle with the most area that can be inscribed in a semicircle of radius r. Show, in fact, that the area of that rectangle is r^2. Let x be the base

Algebra ->  Circles -> SOLUTION: Find the dimensions of the rectangle with the most area that can be inscribed in a semicircle of radius r. Show, in fact, that the area of that rectangle is r^2. Let x be the base       Log On


   

Question 1137441: Find the dimensions of the rectangle with the most area that can be inscribed in a semicircle of radius r. Show, in fact, that the area of that rectangle is r^2. Let x be the base of the rectangle, and let y be its height.
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Allow me to use a different single variable; the suggested ones are not pleasant.

Let the base of the rectangle be 2x. Obviously the center of the semicircle is the midpoint of the base of the rectangle.

Then the height of the rectangle is sqrt%28r%5E2-x%5E2%29.

The the area of the rectangle is base times height: A+=+%282x%29%2A%28sqrt%28r%5E2-x%5E2%29%29

To maximize the area, we find the derivative of the area function using the product rule....



...and find where the derivative is 0 (where the numerator is 0):

2r%5E2-4x%5E2+=+0
r%5E2+=+2x%5E2

The maximum area of the rectangle is then