Answer :
To find the width of the prism, we will go through the following steps:
### Step 1: Find the volume of the rectangular prism
Given:
- Height (h) = 8 meters
- Base area (A) = 3 square meters (as provided/considered example)
The volume [tex]\( V \)[/tex] of a rectangular prism is given by the formula:
[tex]\[ V_{\text{prism}} = \text{base area} \times \text{height} \][/tex]
So,
[tex]\[ V_{\text{prism}} = 3 \, \text{m}^2 \times 8 \, \text{m} = 24 \, \text{m}^3 \][/tex]
Hence, the volume of the rectangular prism is:
[tex]\[ V_{\text{prism}} = 24 \, \text{m}^3 \][/tex]
### Step 2: Find the volume of the cylinder
Given:
- Height (h) = 8 meters
- Let the radius of the cylinder (r) be in meters
The volume [tex]\( V \)[/tex] of a cylinder is given by the formula:
[tex]\[ V_{\text{cylinder}} = \pi \times r^2 \times \text{height} \][/tex]
So,
[tex]\[ V_{\text{cylinder}} = \pi \times r^2 \times 8 \][/tex]
Given that the volume of the rectangular prism is:
[tex]\[ V_{\text{prism}} = 24 \, \text{m}^3 \][/tex]
### Step 3: Set the volumes equal to each other and solve for [tex]\( r \)[/tex]
Since the volumes are equal:
[tex]\[ 24 \, \text{m}^3 = \pi \times r^2 \times 8 \][/tex]
Solving for [tex]\( r^2 \)[/tex]:
[tex]\[ 24 = 8\pi r^2 \][/tex]
[tex]\[ 3 = \pi r^2 \][/tex]
[tex]\[ r^2 = \frac{3}{\pi} \][/tex]
[tex]\[ r = \sqrt{\frac{3}{\pi}} \][/tex]
Given that the calculation yields:
[tex]\[ r \approx 0.977 \text{ meters} \][/tex]
### Final Answer
Hence, the radius [tex]\( r \)[/tex] of the cylinder is approximately:
[tex]\[ r \approx 1.0 \, \text{meters} \][/tex] to the nearest tenth.
### Step 1: Find the volume of the rectangular prism
Given:
- Height (h) = 8 meters
- Base area (A) = 3 square meters (as provided/considered example)
The volume [tex]\( V \)[/tex] of a rectangular prism is given by the formula:
[tex]\[ V_{\text{prism}} = \text{base area} \times \text{height} \][/tex]
So,
[tex]\[ V_{\text{prism}} = 3 \, \text{m}^2 \times 8 \, \text{m} = 24 \, \text{m}^3 \][/tex]
Hence, the volume of the rectangular prism is:
[tex]\[ V_{\text{prism}} = 24 \, \text{m}^3 \][/tex]
### Step 2: Find the volume of the cylinder
Given:
- Height (h) = 8 meters
- Let the radius of the cylinder (r) be in meters
The volume [tex]\( V \)[/tex] of a cylinder is given by the formula:
[tex]\[ V_{\text{cylinder}} = \pi \times r^2 \times \text{height} \][/tex]
So,
[tex]\[ V_{\text{cylinder}} = \pi \times r^2 \times 8 \][/tex]
Given that the volume of the rectangular prism is:
[tex]\[ V_{\text{prism}} = 24 \, \text{m}^3 \][/tex]
### Step 3: Set the volumes equal to each other and solve for [tex]\( r \)[/tex]
Since the volumes are equal:
[tex]\[ 24 \, \text{m}^3 = \pi \times r^2 \times 8 \][/tex]
Solving for [tex]\( r^2 \)[/tex]:
[tex]\[ 24 = 8\pi r^2 \][/tex]
[tex]\[ 3 = \pi r^2 \][/tex]
[tex]\[ r^2 = \frac{3}{\pi} \][/tex]
[tex]\[ r = \sqrt{\frac{3}{\pi}} \][/tex]
Given that the calculation yields:
[tex]\[ r \approx 0.977 \text{ meters} \][/tex]
### Final Answer
Hence, the radius [tex]\( r \)[/tex] of the cylinder is approximately:
[tex]\[ r \approx 1.0 \, \text{meters} \][/tex] to the nearest tenth.
