Chapter 1.7, Problem 68PS

Solution

To find: Thickness of the pencil holder.

The thickness of the pencil holder is equal to $1\text{centimeter}$ .

Given information:

A lump of clay has a volume equal to $200$ cubic centimeters and are asked to make a cylindrical pencil holder. The pencil holder should be $9$ centimeters high and have an inner radius of $3$ centimeters.

  

Calculation:

Consider the following figure.

  

According to the given data,

  $\begin{array}{c}\pi {\left(3+x\right)}^2\left(9\right)-\pi \left(3^2\right)\left(9-x\right)=200\\ 9\pi \left\{{\left(3-x\right)}^2-3^2\right\}=200\\ 9\pi x\left(x+6\right)=200\\ x^2+6x=\frac{200}{9\pi }\end{array}$

Simplify it further as follows.

  $\begin{array}{c}{x}^2+6x+{\left(\frac{6}{2}\right)}^2=\frac{200}{9\pi }+{\left(\frac{6}{2}\right)}^2\\ x^2+6x+9=\frac{81\pi +200}{9\pi }\\ {\left(x+3\right)}^2=\frac{81\pi +200}{9\pi }\\ x=-3+\sqrt{\frac{81\pi +200}{9\pi }}\end{array}$

So, $x\approx 1\text{centimeter}$ that is thickness of the pencil holder is equal to $1\text{centimeter}$ .