Q. 10 - Precalculus Enhanced with Graphing Utilities

Question

A circle of radius r is in inscribed in a square. See the figure.

Part (a): Express the area A of the square as a function of the radius r of the circle.

Part (b): Express the perimeter p of the square as a function of the radius r of the circle.

Step-by-Step Solution

Answer

Part (a): Area of the square as a function is $A\left(x\right)=4r^2$.

Part (a): Perimeter of the square as a function is $p\left(x\right)=8r$.

Part (a) Step 1. Given information.

Consider the given question,

Side of square$\left(a\right)=2r$.

We know the area of the square is $A=a^2$.

Substitute the values,

$$\begin{gathered} A\left(x\right)={\left(2r\right)}^2 \\ A\left(x\right)=4r^2 \end{gathered}$$

Part (b) Step 1. Given information.

Consider the given question,

Side of square$\left(a\right)=2r$

We know the perimeter of the square is $p=4a$.

Substitute the values,

$$\begin{gathered} p\left(x\right)=4\left(2r\right) \\ p\left(x\right)=8r \end{gathered}$$