Expert: Paul Klarreich - 3/18/2011

Question
determine the area of the largest rectangle that can be inscribed inside a semicircle with a radius of ten units. place the length of the rectangle along the diameter.
- i already tried using the the area of a semicircle to help find the area of the rectangle in the semicircle and it didn't wok therefore i am stuck

Answer
Questioner: tori
Country: Canada
Category: Calculus
Private: Yes   <<<<<<<<<<<<<<<<<< changed.
Subject: calculus and vectors- unit 3: derivatives and there applications-chapter 3.3: optimiuzation problems
Question: determine the area of the largest rectangle that can be inscribed inside a semicircle with a radius of ten units. place the length of the rectangle along the diameter.
- i already tried using the the area of a semicircle to help find the area of the rectangle in the semicircle and it didn't wok therefore i am stuck
............................................
Draw your semicircle as:

y = sqrt(100 - x^2)

-- that is half a circle x^2 + y^2 = 10^2, with center at (0,0).

Let x be a point in [0..10].  Then your rectangle has these corners:

(x,0)
(x,y)
(-x,0)
(-x,y)

And you will find that:

base = 2x
height = y.

A = 2xy,

but y = sqrt(100 - x^2)

So maximize

A = 2x sqrt(100 - x^2)

(Trick:  maximize  A^2.  Then you don't need square roots.)

The rest should be routine.