Cone $X$ Has A Diameter Of 20 Ft And A Height Of 16 Ft. Cone $Y$ Is The Same Height As Cone $X$, But Its Diameter Is Twice The Diameter Of Cone $X$.What Is The Difference In The Volumes Of The Two Cones? Use

Mar 13, 2025 by ADMIN 216 views

**Volume of Cones: A Mathematical Analysis**

Introduction

In this article, we will delve into the world of geometry and explore the concept of volume of cones. Specifically, we will examine the difference in volumes of two cones, $X$ and $Y$, with varying diameters and heights. The mathematical analysis will involve the use of formulas and calculations to determine the volume of each cone and ultimately find the difference between them.

The Formula for the Volume of a Cone

The volume of a cone is given by the formula:

V = \frac{1}{3} \pi r^2 h

where $V$ is the volume, $r$ is the radius of the base, and $h$ is the height of the cone.

Cone $X$: A Given Cone

Cone $X$ has a diameter of 20 ft and a height of 16 ft. To find the volume of cone $X$, we need to calculate its radius. Since the diameter is twice the radius, we can find the radius by dividing the diameter by 2:

r_X = \frac{20}{2} = 10 \text{ ft}

Now, we can plug in the values into the formula to find the volume of cone $X$:

V_X = \frac{1}{3} \pi (10)^2 (16)

V_X = \frac{1}{3} \pi (100) (16)

V_X = \frac{1600}{3} \pi \text{ ft}^3

Cone $Y$: A Cone with Twice the Diameter

Cone $Y$ has the same height as cone $X$, but its diameter is twice the diameter of cone $X$. This means that the diameter of cone $Y$ is 40 ft. To find the radius of cone $Y$, we can divide the diameter by 2:

r_Y = \frac{40}{2} = 20 \text{ ft}

Now, we can plug in the values into the formula to find the volume of cone $Y$:

V_Y = \frac{1}{3} \pi (20)^2 (16)

V_Y = \frac{1}{3} \pi (400) (16)

V_Y = \frac{6400}{3} \pi \text{ ft}^3

The Difference in Volumes

To find the difference in volumes of the two cones, we can subtract the volume of cone $X$ from the volume of cone $Y$:

V_Y - V_X = \frac{6400}{3} \pi - \frac{1600}{3} \pi

V_Y - V_X = \frac{4800}{3} \pi

V_Y - V_X = 1600 \pi \text{ ft}^3

Conclusion

In this article, we analyzed the volumes of two cones, $X$ and $Y$, with varying diameters and heights. We used the formula for the volume of a cone to calculate the volume of each cone and ultimately found the difference between them. The result shows that the volume of cone $Y$ is 1600 times the volume of cone $X$.

Mathematical Analysis: A Step-by-Step Guide

Step 1: Calculate the Radius of Cone $X$

To find the radius of cone $X$, we can divide the diameter by 2:

r_X = \frac{20}{2} = 10 \text{ ft}

Step 2: Calculate the Volume of Cone $X$

Now, we can plug in the values into the formula to find the volume of cone $X$:

V_X = \frac{1}{3} \pi (10)^2 (16)

V_X = \frac{1}{3} \pi (100) (16)

V_X = \frac{1600}{3} \pi \text{ ft}^3

Step 3: Calculate the Radius of Cone $Y$

To find the radius of cone $Y$, we can divide the diameter by 2:

r_Y = \frac{40}{2} = 20 \text{ ft}

Step 4: Calculate the Volume of Cone $Y$

Now, we can plug in the values into the formula to find the volume of cone $Y$:

V_Y = \frac{1}{3} \pi (20)^2 (16)

V_Y = \frac{1}{3} \pi (400) (16)

V_Y = \frac{6400}{3} \pi \text{ ft}^3

Step 5: Calculate the Difference in Volumes

To find the difference in volumes of the two cones, we can subtract the volume of cone $X$ from the volume of cone $Y$:

V_Y - V_X = \frac{6400}{3} \pi - \frac{1600}{3} \pi

V_Y - V_X = \frac{4800}{3} \pi

V_Y - V_X = 1600 \pi \text{ ft}^3

Mathematical Formulas and Equations

Formula for the Volume of a Cone

V = \frac{1}{3} \pi r^2 h

Calculations for Cone $X$

r_X = \frac{20}{2} = 10 \text{ ft}

V_X = \frac{1}{3} \pi (10)^2 (16)

V_X = \frac{1600}{3} \pi \text{ ft}^3

Calculations for Cone $Y$

r_Y = \frac{40}{2} = 20 \text{ ft}

V_Y = \frac{1}{3} \pi (20)^2 (16)

V_Y = \frac{6400}{3} \pi \text{ ft}^3

Difference in Volumes

V_Y - V_X = \frac{6400}{3} \pi - \frac{1600}{3} \pi

V_Y - V_X = \frac{4800}{3} \pi

V_Y - V_X = 1600 \pi \text{ ft}^3$<br/> **Q&A: Volume of Cones** =========================

Frequently Asked Questions

Q: What is the formula for the volume of a cone?

A: The formula for the volume of a cone is: