To solve this problem, we need to calculate the volumes of both the original (smaller) cylinder and the larger cylinder (with the radius quadrupled) and then determine how many times greater the larger cylinder's volume is than the smaller cylinder's volume.
Step 1: Calculate the Volume of the Original Cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where:
- \( r \) is the radius of the cylinder,
- \( h \) is the height of the cylinder.
For the smaller cylinder:
- Radius \( r = 3 \) m
- Height \( h = 8 \) m
Substituting in the values:
\[ V_{\text{small}} = \pi (3^2)(8) = \pi (9)(8) = 72\pi \text{ cubic meters} \]
Step 2: Calculate the Volume of the Larger Cylinder
For the larger cylinder where the radius is quadrupled:
- New radius \( r = 4 \times 3 = 12 \) m
- Height remains the same at \( h = 8 \) m
Calculating the volume:
\[ V_{\text{large}} = \pi (12^2)(8) = \pi (144)(8) = 1152\pi \text{ cubic meters} \]
Step 3: Compare the Volumes
Now, we compare the volume of the larger cylinder to the smaller cylinder:
\[ \text{Volume ratio} = \frac{V_{\text{large}}}{V_{\text{small}}} = \frac{1152\pi}{72\pi} \]
The \( \pi \) cancels out:
\[ \text{Volume ratio} = \frac{1152}{72} = 16 \]
Conclusion for Part 1
The volume of the larger cylinder is 16 times greater than the smaller cylinder.
Step 4: How the Volume Changes if Height is Quadrupled
If the height of the larger cylinder is also quadrupled, the new height \( h' = 4 \times 8 = 32 \) m.
Now calculating the new volume with height quadrupled:
\[ V_{\text{new large}} = \pi (12^2)(32) = \pi (144)(32) = 4608\pi \text{ cubic meters} \]
Now, let's find the new volume ratio of the larger cylinder (with both radius and height quadrupled) to the small cylinder:
\[ \text{New volume ratio} = \frac{V_{\text{new large}}}{V_{\text{small}}} = \frac{4608\pi}{72\pi} \] Cancelling \( \pi \):
\[ \text{New volume ratio} = \frac{4608}{72} = 64 \]
Conclusion for Part 2
If both the radius and height of the cylinder are quadrupled, the volume of the larger cylinder becomes 64 times greater than the volume of the smaller cylinder.