Chapter 12.4, Problem 28WE

a.

To determine

To find: the exact volume of rubber.

Answer to Problem 28WE

The volume of hollow sphere is $1382.32{\text{ cm}}^3.$

Explanation of Solution

Given: A hollow rubber ball with inner and outer radius $10$ cm and $11$ cm respectively.

  

Calculation:

Calculate the exact volume of hollow sphere.

  $\begin{array}{l}\text{Volume of sphere}=\frac{4}{3}\pi \left(R^3-r^3\right)\\ =\frac{4}{3}\pi \left({11}^3-{10}^3\right)\\ =\frac{4}{3}\times 3.14\times \left(1331-1000\right)\\ =1.33\times 3.14\times 331\end{array}$

  $=1382.32{\text{ cm}}^3$

Therefore, the volume of hollow sphere is $1382.32{\text{ cm}}^3.$

b.

To determine

To find: the volume of rubber by given formula.

Answer to Problem 28WE

The volume of rubber by given formula is $1256c{m}^3$ .

Explanation of Solution

Given:

Formula to calculate the volume of rubber:

  $V=\left(\text{inner surface area}\right)\times \text{thickness of rubber}$

Calculation:

Inner surface area of hollow sphere:

  $\begin{array}{l}=4\pi r^2\\ =4\pi {\left(10\right)}^2\\ =400\pi c{m}^2\end{array}$

Thickness of rubber:

  $\begin{array}{l}=\left(11-10\right)cm\\ =1cm\end{array}$

Volume of hollow rubber:

  $\begin{array}{l}V=400\pi \times 1\\ =400\pi \\ =400\times 3.14\\ =1256c{m}^3\end{array}$

Comparing the calculation of volume of hollow sphere:

  $\begin{array}{l}\text{Difference in volume}=1382.32-1256\\ =126.32c{m}^3\end{array}$

Therefore, the volume of rubber by given formula is $1256c{m}^3$ .

c.

To determine

To state: that approximation method is better for thick layer of ball or thin layer of ball.

Answer to Problem 28WE

The approximation method is better for thin layer of ball.

Explanation of Solution

Volume of rubber ball calculated by exact formula $=1382.32c{m}^3$

Volume of rubber ball calculated by approximation method $=1256c{m}^3$

From the above calculated results, it can be concluded that approximation method is better for thin layer of ball.