Chapter 4.7, Problem 106E

a.

To determine

Find the angle of repose for rock salt.

Answer to Problem 106E

  $\boxed{{\tan }^{-1}\left(0.6471\right)}Or\boxed{0.57}$ radians.

Explanation of Solution

Given information:

Different types of granular substances naturally settle at different angles when stored in cone-shaped piles. This angle $\theta$ is called the angle of repose (see figure). When rock salt is stored in a cone-shaped pile $11$ feet high, the diameter of the pile’s base is about $34$ feet.

Find the angle of repose for rock salt.

Calculation:

Consider that when rock salt is stored in a cone- shaped pile $11$ feet high, the diameter of the pile’s base is about $34$ feet. The angle $\theta$ is the angle of repose as shown in the figure:

Since the diameter of the base is $34$ feet, then, the radius of the base is $\frac{34}{2}=17$ feet and height of the pile is $11$ feet.

Apply the formula of the tangent function in the right-angled triangle containing $\theta$

  $\begin{array}{l}\tan \theta =\frac{perpendicular}{\text{base}}\\ =\frac{11}{17}\\ \theta ={\tan }^{-1}\left(\frac{11}{17}\right)\\ ={\tan }^{-1}\left(0.6471\right)\end{array}$

Use the TI-83 calculator to evaluate the value of the expression,

Set mode to Radian using keystroke $\boxed{MODE}$ as shown below,

  

Use keystrokes $\boxed{{\text{2}}^{\text{nd}}}$ and $\boxed{MODE}$ . A display will appear. There enter the expression to be evaluated using keystrokes $\boxed{{\text{2}}^{\text{nd}}}$

  $\boxed{TAN}$

  $(0.6471)$ and then press $\boxed{ENTER}$ .

  

Result which appear is,

  $\theta =0.5743$

  $=0.57$

Note that the number $0.6471$ lies in the domain of inverse tangent function that is $(-\infty ,\infty )$

Also, note that $=0.57$ lies in the range of the inverse tangent function that is ,

  $\left(\frac{\pi }{2},\frac{\pi }{2}\right)Or\left(-1.57,1.57\right)$

Hence angle of repose is $\boxed{{\tan }^{-1}\left(0.6471\right)}Or\boxed{0.57}$ radians.

b.

To determine

How tall is a pile of rock salt that has a base diameter of $40$ feet?

Answer to Problem 106E

  $\boxed{13feet}$

Explanation of Solution

Given information:

Different types of granular substances naturally settle at different angles when stored in cone-shaped piles. This angle $\theta$ is called the angle of repose (see figure). When rock salt is stored in a cone-shaped pile $11$ feet high, the diameter of the pile’s base is about $34$ feet.

How tall is a pile of rock salt that has a base diameter of $40$ feet?

Calculation:

Consider that when rock salt is stored in a cone- shaped pile $11$ feet high, the diameter of the pile’s base is about $34$ feet. The angle $\theta$ is the angle of repose as shown in the figure:

  

Now the base-diameter of the rock salt is $40$ feet, then the radius of the base $20$ feet.

Let the height of the pile be $h$ feet.

Therefore, the angle of repose is a characteristic of granular substance. Hence, angle of repose remains same for this particular type of granular substance that is rock-salt.

Apply the formula of the tangent function in the right-angled triangle containing $\theta$

  $\tan \theta =\frac{perpendicular}{base}$

  $=\frac{height}{base}$

  $\begin{array}{l}=\frac{h}{20}\\ h=20(\tan \theta )\\ =20\left[\tan \left({\tan }^{-1}\left(\frac{11}{17}\right)\right)\right]\\ =20\times \frac{11}{17}\\ =12.94\end{array}$

Hence, the height of the pile of rock-salt when the diameter of the pile’s base is $40$ feet is $\boxed{12.94}$ feet and this is approximately $\boxed{13feet}$