A cone has a radius of 4 cm and a height of 9 cm. What is the relationship between the volume of this cone and a cylinder with the same radius and height?

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UpStudy AI · Accepted Answer

To understand the relationship between the volume of a cone and a cylinder with the **same radius** and **height**, let's examine their volume formulas:

1. **Volume of a Cylinder**:
   $$
   V_{\text{cylinder}} = \pi r^2 h
   $$
   - $$ r $$ = radius
   - $$ h $$ = height

2. **Volume of a Cone**:
   $$
   V_{\text{cone}} = \frac{1}{3} \pi r^2 h
   $$
   - $$ r $$ = radius
   - $$ h $$ = height

**Comparing the Two:**

- The volume of the **cone** is **one-third** the volume of the **cylinder** when both have the same radius and height.

**Given the Cone’s Dimensions:**

- **Radius ($$ r $$)** = 4 cm
- **Height ($$ h $$)** = 9 cm

**Volume Calculations:**

1. **Cylinder Volume**:
   $$
   V_{\text{cylinder}} = \pi (4)^2 (9) = 144\pi \text{ cm}^3
   $$

2. **Cone Volume**:
   $$
   V_{\text{cone}} = \frac{1}{3} \pi (4)^2 (9) = 48\pi \text{ cm}^3
   $$

**Conclusion:**

The **volume of the cone** is **one-third** the **volume of the cylinder** with the same radius and height.

**Final Answer:**
A cone with given radius and height has one third the volume of a cylinder with that same radius and height.
The volume of the cone is one-third the volume of the cylinder with the same radius and height.

A cone has a radius of 4 cm and a height of 9 cm. What is the relationship between the volume of this cone and a cylinder with the same radius and height?

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