Chapter 9, Problem 3TP

Solution

To identify: The appropriate tool for the task if a certain task requires making a vertical transition with a slope of $\frac{3}{5}$ . The below given table lists several tools used to make vertical transitions from one point to higher point.

Tools	Range of angles
Ramp	$0^{\circ }-{30}^{\circ }$
Fixed steps	${30}^{\circ }-{35}^{\circ }$
Ladder stairs	${35}^{\circ }-{75}^{\circ }$
Portable stairs	${68}^{\circ }-{80}^{\circ }$
Fixed ladder	${75}^{\circ }-{90}^{\circ }$

The ladder stairs are the most appropriate for accomplishing the task.

Given information:

The given table is:

Tools	Range of angles
Ramp	$0^{\circ }-{30}^{\circ }$
Fixed steps	${30}^{\circ }-{35}^{\circ }$
Ladder stairs	${35}^{\circ }-{75}^{\circ }$
Portable stairs	${68}^{\circ }-{80}^{\circ }$
Fixed ladder	${75}^{\circ }-{90}^{\circ }$

Explanation:

It is given that $\frac{3}{5}$ is the slope made by the vertical translation of a particular task. It means that 3 is the rise and 5 is the run.

  

The length of the opposite angle $\theta$ of is 3 and that of the adjacent side is 5. The tangent ratio can be used to connect 3 and 5.

  $\begin{array}{c}\tan \theta =\frac{\text{opposite}}{\text{adjacent}}\\ =\frac{3}{5}\end{array}$

Now solve for $\theta$ ,

  $\begin{array}{c}\theta ={\tan }^{-1}\frac{3}{5}\\ \approx {30.96}^{\circ }\end{array}$

Therefore, the angle of elevation is about ${30.96}^{\circ }$ .

Now observe the table, it can be seen that the recommended range of the angles of elevation for ladder stairs are ${35}^{\circ }-{75}^{\circ }$ . It means the ladder stairs are the most appropriate for accomplishing the task.