Question

A cylinder and a cone are shown. The cylinder has a radius of 10 and an unspecified height. The cone has a radius of 10 and a height of 4. Dashed lines indicate the circular bases of both shapes.


If the cylinder and the cone shown have the same volume, then the radius and height of the cone could be which of these?

(2 points)
Responses

radius of the cone = 15; height of the cone = 4
radius of the cone = 15; height of the cone = 4

radius of the cone = 15; height of the cone = 12
radius of the cone = 15; height of the cone = 12

radius of the cone = 5; height of the cone = 4
radius of the cone = 5; height of the cone = 4

radius of the cone = 5; height of the cone = 12

Answer 1

To determine the radius and height of the cone that could produce the same volume as the cylinder, we first need to establish the formulae for the volumes of both shapes:

1. **Volume of the cylinder (V₁)**:
\[
V₁ = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height. For the cylinder, \( r = 10 \) and \( h \) is unspecified.

Therefore, the volume of the cylinder is:
\[
V₁ = \pi (10)^2 h = 100\pi h
\]

2. **Volume of the cone (V₂)**:
\[
V₂ = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height. For the cone, we will evaluate the options provided.

Next, we set the two volumes equal to find the corresponding dimensions for the cone:
\[
100\pi h_cylinder = \frac{1}{3} \pi r_{cone}^2 h_{cone}
\]
Removing \( \pi \) from both sides:
\[
100 h_cylinder = \frac{1}{3} r_{cone}^2 h_{cone}
\]
Multiplying both sides by 3:
\[
300 h_cylinder = r_{cone}^2 h_{cone}
\]

Now, we'll evaluate each of the options given for the cone's dimensions where the cone has a radius \( r_{cone} = 10 \) and height \( h_{cone} = 4 \).

**Options:**
1. **Option 1: radius of the cone = 15; height of the cone = 4**
\[
r_{cone}^2 = 15^2 = 225
\]
\[
V_{cone} = \frac{1}{3} \pi (15)^2 (4) = \frac{1}{3} \pi (225)(4) = 300\pi
\]

2. **Option 2: radius of the cone = 15; height of the cone = 12**
\[
r_{cone}^2 = 15^2 = 225
\]
\[
V_{cone} = \frac{1}{3} \pi (15)^2 (12) = \frac{1}{3} \pi (225)(12) = 900\pi
\]

3. **Option 3: radius of the cone = 5; height of the cone = 4**
\[
r_{cone}^2 = 5^2 = 25
\]
\[
V_{cone} = \frac{1}{3} \pi (5)^2 (4) = \frac{1}{3} \pi (25)(4) = \frac{100}{3}\pi
\]

4. **Option 4: radius of the cone = 5; height of the cone = 12**
\[
r_{cone}^2 = 5^2 = 25
\]
\[
V_{cone} = \frac{1}{3} \pi (5)^2 (12) = \frac{1}{3} \pi (25)(12) = 100\pi
\]

Now we need to find which volume is equal to the cylinder:
- The cylinder has a volume of \( 100\pi h \).
- From our calculations:
1. \( 300\pi \) (Option 1)
2. \( 900\pi \) (Option 2)
3. \( \frac{100}{3}\pi \) (Option 3)
4. \( 100\pi \) (Option 4)

None of these volumes matches the cylinder's volume because we need to find \( h_cylinder \).

After comparing, **Option 1** (Volume of cone \( = 300\pi \)) can match if we use \( h_{cylinder} = 1 \) (not specified), since:
\[
100\pi h = 300\pi \text{ implies } h = 3.
\]

The cone with **radius of 15 and height of 4** can indeed produce the equivalent volume.

Thus, the final answer is:
**radius of the cone = 15; height of the cone = 4.**