Chapter 8.6, Problem 42PPSE

Solution

a.

To identify:The equation for $L$

A tapered cylinder is made by decreasing the radius of a rod continuously as you move from one end to the other. The rate at which it tapers is the taper per foot. You can calculate the taper per foot using the formula $T\text{ =}\frac{24\left(R-r\right)}{L}$

The lengths R, r, and L are measured in inches.

  

The equation for $\text{L =}\frac{24\left(R-r\right)}{T}$

Given information:

The given is the formula $T\text{ =}\frac{24\left(R-r\right)}{L}$

The lengths R, r, and L are measured in inches.

  

Explanation:

Consider the given information

The equation for $L$ is.

  $\text{L =}\frac{24\left(R-r\right)}{T}$

Hence the equation for $\text{L =}\frac{24\left(R-r\right)}{T}$

b.

To identify:What is L for $T\text{ =}0.75,0.85,\text{ and }0.95,i\text{f }R=4in.;r=3in.$

The value of $L\text{for}\text{T=0}\text{.75}$ is $\text{32 inches}$ .

The value of $L\text{for}\text{T=0}\text{.85}$ is $\text{28}\text{.2 inches}$ .

The value of $L\text{for}\text{T=0}\text{.95}$ is $\text{25}\text{.3 inches}$ .

Given information:

The given is the formula for $T\text{ =}\frac{24\left(R-r\right)}{L}$

The lengths R, r, and L are measured in inches.

  

Explanation:

Consider the given information.

Substitute $R=4,r=3,T=0.75$ in the result of (a) then

  $\begin{array}{c}\text{L =}\frac{24\left(R-r\right)}{T}\\ L=\frac{24\left(4-3\right)}{0.75}\\ L=\frac{24}{0.75}\\ L=32\end{array}$

Hence the value of $L\text{for}\text{T=0}\text{.75}$ is $\text{32 inches}$

Substitute $R=4,r=3,T=0.85$ in the result of (a) then

  $\begin{array}{c}\text{L =}\frac{24\left(R-r\right)}{T}\\ L=\frac{24\left(4-3\right)}{0.85}\\ L=\frac{24}{0.85}\\ L\approx 28.2\end{array}$

Hence the value of $L\text{for}\text{T=0}\text{.85}$ is $\text{28}\text{.2 inches}$

Substitute $R=4,r=3,T=0.95$ in the result of (a) then

  $\begin{array}{c}\text{L =}\frac{24\left(R-r\right)}{T}\\ L=\frac{24\left(4-3\right)}{0.95}\\ L=\frac{24}{0.95}\\ L\approx 25.3\end{array}$

Hence the value of $L\text{for}\text{T=0}\text{.95}$ is $\text{25}\text{.3 inches}$