Rectangle Inscribed in Semicircle

**symmetry** · January 22nd 2007, 05:37 PM

A rectangle is Inscribed in a semicircle of radius 2.
Let P = (x, y) be the point in quadrant 1 that is a vertex of the rectangle and is on the circle.

(a) Express the area A of the rectangle as a function of x.

(b) Express the perimeter p of the rectangle as a function of x.

**earboth** · January 22nd 2007, 10:06 PM

Originally Posted by symmetry

A rectangle is Inscribed in a semicircle of radius 2.
Let P = (x, y) be the point in quadrant 1 that is a vertex of the rectangle and is on the circle.

(a) Express the area A of the rectangle as a function of x.

(b) Express the perimeter p of the rectangle as a function of x.

Hello,

to (a)

The area of the rectangle is calculated in general by:

$A=length \times width$

According to your problem the area is:

$A=2x \cdot y$ . Now use Pythagoran theorem:

$x^2+y^2=r^2 \Longleftrightarrow y=\sqrt{r^2-x^2}$ . Thus:

$A=2x \cdot \sqrt{r^2-x^2}$ . That means with r = 2:

$A=2x \cdot \sqrt{4-x^2}$

to (b):

The perimeter of a rectangle is calculated in general:

$p=2 \cdot {length} + 2 \cdot {width}$ . Plug in the values you know:

$p=2 \cdot 2x + 2 \cdot \sqrt{4-x^2}= 4x+ 2 \cdot \sqrt{4-x^2}$