Question
Find the volume of each cylinder. Which cylinder has the greater volume? Use 3.14 for π and round your answers to the nearest hundredth.
Two cylinders. The circumference of the base of cylinder A is 3 meters and the height of the cylinder is 5 meters. The circumference of the base of cylinder B is 5 meters and the height of the cylinder is 3 meters.
Cylinder A has a volume of
m3, and Cylinder B has a volume of
m3. Cylinder
has the greater volume.
Two cylinders. The circumference of the base of cylinder A is 3 meters and the height of the cylinder is 5 meters. The circumference of the base of cylinder B is 5 meters and the height of the cylinder is 3 meters.
Cylinder A has a volume of
m3, and Cylinder B has a volume of
m3. Cylinder
has the greater volume.
Answers
Answered by
GPT-4o mini
To find the volume of each cylinder, we will use the formula for the volume of a cylinder:
\[
V = \pi r^2 h
\]
where \( V \) is the volume, \( r \) is the radius of the base, and \( h \) is the height of the cylinder.
### Step 1: Find the radius of Cylinder A
The circumference of Cylinder A is given as 3 meters. The formula for the circumference is:
\[
C = 2\pi r
\]
Rearranging this gives:
\[
r = \frac{C}{2\pi} = \frac{3}{2 \times 3.14} \approx \frac{3}{6.28} \approx 0.477 \text{ meters}
\]
### Step 2: Calculate the volume of Cylinder A
With the radius \( r \approx 0.477 \) meters and height \( h = 5 \) meters:
\[
V_A = \pi r^2 h = 3.14 \times (0.477)^2 \times 5
\]
Calculating \( (0.477)^2 \):
\[
(0.477)^2 \approx 0.227729
\]
Now substituting this back into the volume formula:
\[
V_A \approx 3.14 \times 0.227729 \times 5 \approx 3.14 \times 1.138645 \approx 3.57 \text{ m}^3
\]
### Step 3: Find the radius of Cylinder B
The circumference of Cylinder B is given as 5 meters. Using the same formula:
\[
r = \frac{C}{2\pi} = \frac{5}{2 \times 3.14} \approx \frac{5}{6.28} \approx 0.797 \text{ meters}
\]
### Step 4: Calculate the volume of Cylinder B
With radius \( r \approx 0.797 \) meters and height \( h = 3 \) meters:
\[
V_B = \pi r^2 h = 3.14 \times (0.797)^2 \times 3
\]
Calculating \( (0.797)^2 \):
\[
(0.797)^2 \approx 0.635209
\]
Now substituting this back into the volume formula:
\[
V_B \approx 3.14 \times 0.635209 \times 3 \approx 3.14 \times 1.905627 \approx 5.98 \text{ m}^3
\]
### Conclusion
The volumes of the cylinders are:
- **Cylinder A**: \( \approx 3.57 \text{ m}^3 \)
- **Cylinder B**: \( \approx 5.98 \text{ m}^3 \)
Therefore, Cylinder B has the greater volume.
Final results:
- Cylinder A has a volume of **3.57 m³**
- Cylinder B has a volume of **5.98 m³**
- Cylinder **B** has the greater volume.
\[
V = \pi r^2 h
\]
where \( V \) is the volume, \( r \) is the radius of the base, and \( h \) is the height of the cylinder.
### Step 1: Find the radius of Cylinder A
The circumference of Cylinder A is given as 3 meters. The formula for the circumference is:
\[
C = 2\pi r
\]
Rearranging this gives:
\[
r = \frac{C}{2\pi} = \frac{3}{2 \times 3.14} \approx \frac{3}{6.28} \approx 0.477 \text{ meters}
\]
### Step 2: Calculate the volume of Cylinder A
With the radius \( r \approx 0.477 \) meters and height \( h = 5 \) meters:
\[
V_A = \pi r^2 h = 3.14 \times (0.477)^2 \times 5
\]
Calculating \( (0.477)^2 \):
\[
(0.477)^2 \approx 0.227729
\]
Now substituting this back into the volume formula:
\[
V_A \approx 3.14 \times 0.227729 \times 5 \approx 3.14 \times 1.138645 \approx 3.57 \text{ m}^3
\]
### Step 3: Find the radius of Cylinder B
The circumference of Cylinder B is given as 5 meters. Using the same formula:
\[
r = \frac{C}{2\pi} = \frac{5}{2 \times 3.14} \approx \frac{5}{6.28} \approx 0.797 \text{ meters}
\]
### Step 4: Calculate the volume of Cylinder B
With radius \( r \approx 0.797 \) meters and height \( h = 3 \) meters:
\[
V_B = \pi r^2 h = 3.14 \times (0.797)^2 \times 3
\]
Calculating \( (0.797)^2 \):
\[
(0.797)^2 \approx 0.635209
\]
Now substituting this back into the volume formula:
\[
V_B \approx 3.14 \times 0.635209 \times 3 \approx 3.14 \times 1.905627 \approx 5.98 \text{ m}^3
\]
### Conclusion
The volumes of the cylinders are:
- **Cylinder A**: \( \approx 3.57 \text{ m}^3 \)
- **Cylinder B**: \( \approx 5.98 \text{ m}^3 \)
Therefore, Cylinder B has the greater volume.
Final results:
- Cylinder A has a volume of **3.57 m³**
- Cylinder B has a volume of **5.98 m³**
- Cylinder **B** has the greater volume.