**Problem Statement: Calculating the Volume of a Cone****Problem:**Find the volume of a cone with a base diameter of 12 yards (yd) and a height of 11 yards (yd). Write the exact volume in terms of π, and be sure to include the correct unit in your answer.**Solution:**To find the volume of a cone, we use the formula:\[ V = \frac{1}{3} \pi r^2 h \]Where:- $ V $ is the volume of the cone- $ r $ is the radius of the base of the cone- $ h $ is the height of the coneGiven:- The diameter of the cone's base is 12 yards. Therefore, the radius $ r $ is half of the diameter: \[ r = \frac{12 \text{ yd}}{2} = 6 \text{ yd} \]- The height $ h $ of the cone is 11 yards.By substituting these values into the formula, we get:\[ V = \frac{1}{3} \pi (6 \text{ yd})^2 (11 \text{ yd}) \]First, calculate the radius squared:\[ (6 \text{ yd})^2 = 36 \text{ yd}^2 \]Next, multiply by the height:\[ 36 \text{ yd}^2 \times 11 \text{ yd} = 396 \text{ yd}^3 \]Finally, multiply by $ \frac{1}{3} $:\[ V = \frac{1}{3} \cdot 396 \pi \text{ yd}^3 = 132 \pi \text{ yd}^3 \]Therefore, the exact volume of the cone is:\[ V = 132 \pi \text{ yd}^3 \] This image illustrates a cone with specific measurements, useful for learning about three-dimensional geometric shapes. The cone is depicted with its vertical height, radius, and slant height. Here are the details provided by the diagram:1. **Height**: The vertical height of the cone is 11 yards (yd).2. **Radius**: The radius of the cone's base is 12 yards (yd).The diagram includes the following features:- A cone shape with a dashed line indicating the height of the cone starting from the apex (top of the cone) to the center of the base. This line measures 11 yards, showing the vertical height of the cone.- A right-angled triangle is formed within the cone by the vertical height (11 yd), the radius (12 yd), and the slant height (hypotenuse of the triangle).- The circular base of the cone is shown with a dashed outline, emphasizing the shape's round base.- There is a small square symbol at the intersection of the height and the radius, indicating a right angle (90 degrees), which confirms that the vertical height is perpendicular to the base.This geometric representation helps in visualizing and understanding the components and properties of a cone, such as its height, radius, and how the slant height can be derived using the Pythagorean Theorem.

Answered: Image /qna-images/answer/93799f48-be2d-4d11-aae9-be0fd7871839.jpg

Answered: yd 12 yd

Math

Geometry

yd 12 yd

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter9: Surfaces And Solids

Section9.2: Pyramids, Area, And Volume

Problem 33E

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Question

**Problem Statement: Calculating the Volume of a Cone**

**Problem:**
Find the volume of a cone with a base diameter of 12 yards (yd) and a height of 11 yards (yd). Write the exact volume in terms of π, and be sure to include the correct unit in your answer.

**Solution:**
To find the volume of a cone, we use the formula:
\[ V = \frac{1}{3} \pi r^2 h \]

Where:
- $ V $ is the volume of the cone
- $ r $ is the radius of the base of the cone
- $ h $ is the height of the cone

Given:
- The diameter of the cone's base is 12 yards. Therefore, the radius $ r $ is half of the diameter:
\[ r = \frac{12 \text{ yd}}{2} = 6 \text{ yd} \]
- The height $ h $ of the cone is 11 yards.

By substituting these values into the formula, we get:
\[ V = \frac{1}{3} \pi (6 \text{ yd})^2 (11 \text{ yd}) \]

First, calculate the radius squared:
\[ (6 \text{ yd})^2 = 36 \text{ yd}^2 \]

Next, multiply by the height:
\[ 36 \text{ yd}^2 \times 11 \text{ yd} = 396 \text{ yd}^3 \]

Finally, multiply by $ \frac{1}{3} $:
\[ V = \frac{1}{3} \cdot 396 \pi \text{ yd}^3 = 132 \pi \text{ yd}^3 \]

Therefore, the exact volume of the cone is:
\[ V = 132 \pi \text{ yd}^3 \]

This image illustrates a cone with specific measurements, useful for learning about three-dimensional geometric shapes. The cone is depicted with its vertical height, radius, and slant height. Here are the details provided by the diagram:

1. **Height**: The vertical height of the cone is 11 yards (yd).
2. **Radius**: The radius of the cone's base is 12 yards (yd).

The diagram includes the following features:

- A cone shape with a dashed line indicating the height of the cone starting from the apex (top of the cone) to the center of the base. This line measures 11 yards, showing the vertical height of the cone.
- A right-angled triangle is formed within the cone by the vertical height (11 yd), the radius (12 yd), and the slant height (hypotenuse of the triangle).
- The circular base of the cone is shown with a dashed outline, emphasizing the shape's round base.
- There is a small square symbol at the intersection of the height and the radius, indicating a right angle (90 degrees), which confirms that the vertical height is perpendicular to the base.

This geometric representation helps in visualizing and understanding the components and properties of a cone, such as its height, radius, and how the slant height can be derived using the Pythagorean Theorem.

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