Question
Aiden evaluated the expression mc011-1.jpg to find the volume of a cylinder. What could be dimensions of the cylinder?
The height is 12 cm, and the diameter of the base is 8 cm.
The height is 12 cm, and the area of the base is 16 cm2.
The height is 16 cm, and the diameter of the base is 6 cm.
The height is 16 cm, and the area of the base is 12 cm2.
The height is 12 cm, and the diameter of the base is 8 cm.
The height is 12 cm, and the area of the base is 16 cm2.
The height is 16 cm, and the diameter of the base is 6 cm.
The height is 16 cm, and the area of the base is 12 cm2.
Answers
GPT-4o mini
To find the volume of a cylinder, you can use the formula:
\[
V = \pi r^2 h
\]
where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.
Given that the diameter is twice the radius (\( d = 2r \)), let's examine each option based on the formula:
1. **The height is 12 cm, and the diameter of the base is 8 cm.**
- Radius \( r = \frac{8}{2} = 4 \) cm
- Volume \( V = \pi (4)^2 (12) = \pi \times 16 \times 12 = 192\pi \) cm³
2. **The height is 12 cm, and the area of the base is 16 cm².**
- Area of the base \( A = \pi r^2 = 16 \) cm²
- To find the radius, we solve for \( r \):
\[
r^2 = \frac{16}{\pi} \implies r = \sqrt{\frac{16}{\pi}} \text{ cm}
\]
- Volume \( V = 16 \times 12 = 192 \) cm³ (since we know the area of the base).
3. **The height is 16 cm, and the diameter of the base is 6 cm.**
- Radius \( r = \frac{6}{2} = 3 \) cm
- Volume \( V = \pi (3)^2 (16) = \pi \times 9 \times 16 = 144\pi \) cm³
4. **The height is 16 cm, and the area of the base is 12 cm².**
- Area of the base \( A = 12 \) cm²
- To find the radius:
\[
r^2 = \frac{12}{\pi} \implies r = \sqrt{\frac{12}{\pi}} \text{ cm}
\]
- Volume \( V = 12 \times 16 = 192 \) cm³ (since we calculate using the area of the base).
Thus, the scenarios for volume are:
- The first, second, and fourth options yield a calculated volume of \( 192 \) cm³.
- The third option yields \( 144\pi \) cm³, which is less than \( 192 \) cm³.
Therefore, the dimensions of the cylinder can be either:
- Height of 12 cm, diameter of 8 cm.
- Height of 12 cm, area of base 16 cm².
- Height of 16 cm, area of base 12 cm².
\[
V = \pi r^2 h
\]
where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.
Given that the diameter is twice the radius (\( d = 2r \)), let's examine each option based on the formula:
1. **The height is 12 cm, and the diameter of the base is 8 cm.**
- Radius \( r = \frac{8}{2} = 4 \) cm
- Volume \( V = \pi (4)^2 (12) = \pi \times 16 \times 12 = 192\pi \) cm³
2. **The height is 12 cm, and the area of the base is 16 cm².**
- Area of the base \( A = \pi r^2 = 16 \) cm²
- To find the radius, we solve for \( r \):
\[
r^2 = \frac{16}{\pi} \implies r = \sqrt{\frac{16}{\pi}} \text{ cm}
\]
- Volume \( V = 16 \times 12 = 192 \) cm³ (since we know the area of the base).
3. **The height is 16 cm, and the diameter of the base is 6 cm.**
- Radius \( r = \frac{6}{2} = 3 \) cm
- Volume \( V = \pi (3)^2 (16) = \pi \times 9 \times 16 = 144\pi \) cm³
4. **The height is 16 cm, and the area of the base is 12 cm².**
- Area of the base \( A = 12 \) cm²
- To find the radius:
\[
r^2 = \frac{12}{\pi} \implies r = \sqrt{\frac{12}{\pi}} \text{ cm}
\]
- Volume \( V = 12 \times 16 = 192 \) cm³ (since we calculate using the area of the base).
Thus, the scenarios for volume are:
- The first, second, and fourth options yield a calculated volume of \( 192 \) cm³.
- The third option yields \( 144\pi \) cm³, which is less than \( 192 \) cm³.
Therefore, the dimensions of the cylinder can be either:
- Height of 12 cm, diameter of 8 cm.
- Height of 12 cm, area of base 16 cm².
- Height of 16 cm, area of base 12 cm².