### Understanding the Angle of Repose in Granular MaterialsWhen granular materials fall, they form conical piles. The angle of slope, measured from the horizontal, at which the material comes to rest is called the **angle of repose**. The angle of repose $ \theta $ is related to the height $ h $ and base radius $ r $ of the conical pile by the equation:\[ \theta = \cot^{-1}\left(\frac{r}{h}\right) \]For instance, if granular materials are stored in a pile 10 feet high and the diameter of the base of the pile is 25 feet, you can determine various aspects of the pile using the provided formula.Below the formula, there are some practical exercises:#### (a) Calculating the Angle of Repose**Find the angle of repose for these granular materials:**\[ \theta = \boxed{\phantom{00}}^\circ \](Round to two decimal places as needed.)#### (b) Determining Base Diameter**What is the base diameter of a pile that is 15 feet high?**\[ \text{diameter} = \boxed{\phantom{000}} \text{ feet} \](Round to two decimal places as needed.)#### (c) Finding the Height of the Pile**What is the height of a pile that has a base diameter of approximately 114 feet?**\[ \text{height} = \boxed{\phantom{000}} \text{ feet} \](Round to two decimal places as needed.)### Visual RepresentationThe image includes a diagram of a conical pile illustrating the relationship between the height ($ h $), radius ($ r $), and angle of repose ($ \theta $). The height is shown as a vertical line from the base to the apex of the cone. The radius is the horizontal distance from the center of the base to its edge. The angle of repose ($ \theta $) is represented as the angle between the height line and the sloped surface of the pile.Understanding these relationships is crucial for fields like material science, civil engineering, and geotechnics, where handling and storage of granular materials are common.

Given angle of repose , θ= cot-1 rha) given h=10 ft , d= 25 ft r=…

When granular materials fall, they form conical piles. The angle of slope, measured from the horizontal, at which the material comes to rest is called the angle of repose. The angle of repose is related to the height h and base radius r of the conical pile by the equation 0 = cot -1 When certain h granular materials are stored in a pile 10 feet high, the diameter of the base of the pile is 25 feet.

When granular materials fall, they form conical piles. The angle of slope, measured from the horizontal, at which the material comes to rest is called the angle of repose. The angle of repose is related to the height h and base radius r of the conical pile by the equation 0 = cot -1 When certain h granular materials are stored in a pile 10 feet high, the diameter of the base of the pile is 25 feet.

Trigonometry (11th Edition)

11th Edition

ISBN:9780134217437

Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Chapter1: Trigonometric Functions

Section: Chapter Questions

Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.

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### Understanding the Angle of Repose in Granular Materials

When granular materials fall, they form conical piles. The angle of slope, measured from the horizontal, at which the material comes to rest is called the **angle of repose**. The angle of repose $ \theta $ is related to the height $ h $ and base radius $ r $ of the conical pile by the equation:

\[ \theta = \cot^{-1}\left(\frac{r}{h}\right) \]

For instance, if granular materials are stored in a pile 10 feet high and the diameter of the base of the pile is 25 feet, you can determine various aspects of the pile using the provided formula.

Below the formula, there are some practical exercises:

#### (a) Calculating the Angle of Repose
**Find the angle of repose for these granular materials:**

\[ \theta = \boxed{\phantom{00}}^\circ \]
(Round to two decimal places as needed.)

#### (b) Determining Base Diameter
**What is the base diameter of a pile that is 15 feet high?**

\[ \text{diameter} = \boxed{\phantom{000}} \text{ feet} \]
(Round to two decimal places as needed.)

#### (c) Finding the Height of the Pile
**What is the height of a pile that has a base diameter of approximately 114 feet?**

\[ \text{height} = \boxed{\phantom{000}} \text{ feet} \]
(Round to two decimal places as needed.)

### Visual Representation

The image includes a diagram of a conical pile illustrating the relationship between the height ($ h $), radius ($ r $), and angle of repose ($ \theta $). The height is shown as a vertical line from the base to the apex of the cone. The radius is the horizontal distance from the center of the base to its edge. The angle of repose ($ \theta $) is represented as the angle between the height line and the sloped surface of the pile.

Understanding these relationships is crucial for fields like material science, civil engineering, and geotechnics, where handling and storage of granular materials are common.

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