Chapter 1.8, Problem 61E

(a)

To determine

Find the radius r of the tank as a function of the length x of the sides of the square

Answer to Problem 61E

The radius r of the tank as a function of the length x of the sides of the square

  $r\left(x\right)=\frac{x}{2}$

Explanation of Solution

Given:

A square concrete foundation is a base for a cylindrical tank.

  

Base radius of the tank,

  $r\left(x\right)=\frac{x}{2}$ , $x$ is the length of the side of the square

(b)

To determine

Find the area A of the circular base of the tank as a function of the radius r.

Answer to Problem 61E

The area A of the circular base of the tank as a function of the radius r

  $A\left(r\right)=\pi r^2$

Explanation of Solution

Given:

A square concrete foundation is a base for a cylindrical tank.

  

Area of the circular base of the tank as a function of the radius r

  $A\left(r\right)=\pi r^2$

(c)

To determine

Find and interpret $(A\circ r)(x)$

Answer to Problem 61E

  $(A\circ r)(x)$

  $=\frac{\pi x^2}{4}$

The quantity $(A\circ r)(x)$ represents the area of the circular base of the tank as function of the length $\left(x\right)$ of the square foundation

Explanation of Solution

Given:

A square concrete foundation is a base for a cylindrical tank.

  

Compute the value of $(A\circ r)(x)$

  $(A\circ r)(x)=A(r(x))$

  $\begin{array}{l}=\pi {\left(r\left(x\right)\right)}^2\hfill \\ =\pi {\left(\frac{x}{2}\right)}^2\hfill \\ =\frac{\pi x^2}{4}\hfill \end{array}$

The quantity $(A\circ r)(x)$ represents the area of the circular base of the tank as function of the length $\left(x\right)$ of the square foundation.