To find out how many times greater the volume of the larger cylinder is compared to the smaller cylinder, we need to first calculate the volumes of both cylinders.
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.
Step 1: Calculate the volume of the smaller cylinder
For the smaller cylinder:
- Radius \( r_1 = 6 \) m
- Height \( h_1 = 12 \) m
Using the volume formula:
\[ V_1 = \pi (6)^2 (12) \]
Calculating that step by step:
\[ V_1 = \pi (36) (12) \] \[ V_1 = \pi (432) \] \[ V_1 = 432\pi \text{ m}^3 \]
Step 2: Calculate the volume of the larger cylinder
For the larger cylinder, the radius is quadrupled:
- New radius \( r_2 = 4 \times 6 = 24 \) m
- Height remains the same \( h_2 = 12 \) m
Calculating its volume:
\[ V_2 = \pi (24)^2 (12) \]
Calculating that step by step:
\[ V_2 = \pi (576) (12) \] \[ V_2 = \pi (6912) \] \[ V_2 = 6912\pi \text{ m}^3 \]
Step 3: Compare the volumes
Now, we want to find out how many times greater \( V_2 \) is than \( V_1 \):
\[ \text{Ratio} = \frac{V_2}{V_1} = \frac{6912\pi}{432\pi} \]
The \( \pi \) cancels out:
\[ \text{Ratio} = \frac{6912}{432} \]
Now perform the division:
\[ \text{Ratio} = 16 \]
Conclusion for the first part
Therefore, the volume of the larger cylinder is 16 times greater than the volume of the smaller cylinder.
Step 4: If the height were also quadrupled
Now let's consider if the height of the larger cylinder is also quadrupled. The new height will be:
\[ h_2' = 4 \times 12 = 48 \text{ m} \]
We can recalculate the volume for the larger cylinder with the new height:
The volume now becomes:
\[ V_2' = \pi (24)^2 (48) \]
Calculating that:
\[ V_2' = \pi (576) (48) = \pi (27648) = 27648\pi \text{ m}^3 \]
Step 5: Calculate the new ratio
Now we compare this new volume \( V_2' \) to the original volume \( V_1 \):
\[ \text{New Ratio} = \frac{V_2'}{V_1} = \frac{27648\pi}{432\pi} \]
The \( \pi \) cancels out again:
\[ \text{New Ratio} = \frac{27648}{432} \]
Performing the division:
\[ \text{New Ratio} = 64 \]
Conclusion for the second part
If both the radius and the height were quadrupled, the volume of the larger cylinder would be 64 times greater than the volume of the smaller cylinder.