The Volume Of A Cylinder Is $156 \pi , \text{cm}^3$. What Is The Volume Of A Cone With The Same Base And Height As The Cylinder?A. $39 \pi , \text{cm}^3$ B. $ 52 Π Cm 3 52 \pi \, \text{cm}^3 52 Π Cm 3 [/tex] C. $78 \pi ,

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Introduction

In geometry, the volume of a three-dimensional shape is a measure of the amount of space inside the shape. When comparing the volumes of different shapes, it's essential to understand the formulas and relationships between their dimensions. In this article, we'll explore the relationship between the volume of a cylinder and a cone with the same base and height.

The Volume of a Cylinder

The volume of a cylinder is given by the formula:

Vcylinder=πr2hV_{cylinder} = \pi r^2 h

where rr is the radius of the base and hh is the height of the cylinder. In this case, the volume of the cylinder is given as $156 \pi , \text{cm}^3$.

The Volume of a Cone

The volume of a cone is given by the formula:

Vcone=13πr2hV_{cone} = \frac{1}{3} \pi r^2 h

where rr is the radius of the base and hh is the height of the cone. Since the cone has the same base and height as the cylinder, we can substitute the values of rr and hh from the cylinder into the formula for the cone.

Calculating the Volume of the Cone

Let's calculate the volume of the cone using the formula:

Vcone=13πr2hV_{cone} = \frac{1}{3} \pi r^2 h

We know that the volume of the cylinder is $156 \pi , \text{cm}^3$, and the formula for the volume of a cylinder is:

Vcylinder=πr2hV_{cylinder} = \pi r^2 h

We can set up a proportion to relate the volumes of the cylinder and the cone:

VcylinderVcone=πr2h13πr2h\frac{V_{cylinder}}{V_{cone}} = \frac{\pi r^2 h}{\frac{1}{3} \pi r^2 h}

Simplifying the proportion, we get:

VcylinderVcone=3\frac{V_{cylinder}}{V_{cone}} = 3

Now, we can substitute the value of the volume of the cylinder into the proportion:

156π cm3Vcone=3\frac{156 \pi \, \text{cm}^3}{V_{cone}} = 3

Solving for the volume of the cone, we get:

Vcone=156π cm33V_{cone} = \frac{156 \pi \, \text{cm}^3}{3}

Vcone=52π cm3V_{cone} = 52 \pi \, \text{cm}^3

Conclusion

In conclusion, the volume of a cone with the same base and height as a cylinder is one-third the volume of the cylinder. Therefore, the volume of the cone is $52 \pi , \text{cm}^3$.

Answer

The correct answer is:

  • B. $52 \pi , \text{cm}^3$

References

  • [1] "Geometry Formulas". Math Open Reference. Retrieved 2023-12-15.
  • [2] "Cylinder and Cone Volume Formulas". Math Is Fun. Retrieved 2023-12-15.

Related Topics

Introduction

In our previous article, we explored the relationship between the volume of a cylinder and a cone with the same base and height. We calculated that the volume of the cone is one-third the volume of the cylinder. In this article, we'll answer some frequently asked questions related to the volume of a cone and a cylinder.

Q&A

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

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

Vcone=13πr2hV_{cone} = \frac{1}{3} \pi r^2 h

where rr is the radius of the base and hh is the height of the cone.

Q: What is the relationship between the volume of a cylinder and a cone with the same base and height?

A: The volume of a cone with the same base and height as a cylinder is one-third the volume of the cylinder.

Q: How do I calculate the volume of a cone?

A: To calculate the volume of a cone, you need to know the radius of the base and the height of the cone. You can use the formula:

Vcone=13πr2hV_{cone} = \frac{1}{3} \pi r^2 h

Q: What is the volume of a cone with a radius of 4 cm and a height of 6 cm?

A: To calculate the volume of the cone, we need to plug in the values of the radius and height into the formula:

Vcone=13π(4)2(6)V_{cone} = \frac{1}{3} \pi (4)^2 (6)

Vcone=13π(16)(6)V_{cone} = \frac{1}{3} \pi (16) (6)

Vcone=13π(96)V_{cone} = \frac{1}{3} \pi (96)

Vcone=32π cm3V_{cone} = 32 \pi \, \text{cm}^3

Q: What is the volume of a cylinder with a radius of 4 cm and a height of 6 cm?

A: To calculate the volume of the cylinder, we need to plug in the values of the radius and height into the formula:

Vcylinder=π(4)2(6)V_{cylinder} = \pi (4)^2 (6)

Vcylinder=π(16)(6)V_{cylinder} = \pi (16) (6)

Vcylinder=π(96)V_{cylinder} = \pi (96)

Vcylinder=96π cm3V_{cylinder} = 96 \pi \, \text{cm}^3

Q: How does the volume of a cone compare to the volume of a cylinder?

A: The volume of a cone is one-third the volume of a cylinder with the same base and height.

Q: What is the relationship between the radius and height of a cone and its volume?

A: The volume of a cone is directly proportional to the square of the radius and the height.

Q: Can I use the formula for the volume of a cone to calculate the volume of a cylinder?

A: No, the formula for the volume of a cone is not the same as the formula for the volume of a cylinder. You need to use the correct formula for the shape you are calculating the volume for.

Conclusion

In conclusion, we've answered some frequently asked questions related to the volume of a cone and a cylinder. We've explored the relationship between the volume of a cylinder and a cone with the same base and height, and we've calculated the volume of a cone with a given radius and height.

References

  • [1] "Geometry Formulas". Math Open Reference. Retrieved 2023-12-15.
  • [2] "Cylinder and Cone Volume Formulas". Math Is Fun. Retrieved 2023-12-15.

Related Topics