Q. 15 - Precalculus Enhanced with Graphing Utilities

Question

A semicircle of radius r is inscribed in a rectangle so that the diameter of the semicircle is the length of the rectangle. See the figure.

(a) Express the area A of the rectangle as a function of the radius r of the semicircle.

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

Step-by-Step Solution

Answer

Part (a) Area A of rectangle as a function of the radius, r of the semicircle is $2r^2$.

Part (b) The perimeter p of the rectangle as a function of r is $6r$.

Part (a) Step 1. Given

The length of the rectangle equals the diameter of the semicircle$=2r$

The breadth of the rectangle equals the radius of the semicircle$=r$

Part (a) Step 2. Substitute given values in the formula of the area of the rectangle.

Area A of the rectangle $=lb$

$$\begin{gathered} A=lb \\ =2r\left(r\right) \\ =2r^2 \end{gathered}$$

Area of the rectangle as a function of the radius of the semicircle is $2r^2$.

Part (b) Step 1. Substitute the give values of length and breadth of the rectangle in the formula for the perimeter of the rectangle.

Length of the rectangle$=2r$

Breadth of the rectangle$=r$

Perimeter p of the rectangle$=2(l+b)$

$$\begin{gathered} p=2(l+b) \\ p=2(2r+r) \\ p=2\left(3r\right) \\ p=6r \end{gathered}$$

The perimeter of the rectangle as a function of the radius of the semicircle is 6r.