To find how many times greater the volume of the larger cylinder is compared to the smaller cylinder, we need to start by calculating the volumes of both cylinders using the formula for the volume of a cylinder:
\[ V = \pi r^2 h \]
Volume of the Smaller Cylinder
Given:
- Radius \( r = 9 \) cm
- Height \( h = 6 \) cm
Calculating the volume:
\[ V_{\text{small}} = \pi (9^2)(6) = \pi (81)(6) = 486\pi \text{ cm}^3 \]
Volume of the Larger Cylinder
With the radius doubled:
- New radius \( r = 2 \times 9 \) cm = 18 cm
- Height remains the same \( h = 6 \) cm
Calculating the volume of the larger cylinder:
\[ V_{\text{large}} = \pi (18^2)(6) = \pi (324)(6) = 1944\pi \text{ cm}^3 \]
Comparing the Two Volumes
To find how many times greater the volume of the larger cylinder is than that of the smaller cylinder, we can divide the volume of the larger cylinder by the volume of the smaller cylinder:
\[ \text{Ratio} = \frac{V_{\text{large}}}{V_{\text{small}}} = \frac{1944\pi}{486\pi} \]
The \(\pi\) cancels out:
\[ \text{Ratio} = \frac{1944}{486} = 4 \]
Thus, the volume of the larger cylinder is 4 times greater than the volume of the smaller cylinder.
Effect of Doubling the Height
Now, let’s consider the scenario where both the height and the radius are doubled:
- New radius \( r = 18 \) cm (already calculated)
- New height \( h = 2 \times 6 \) cm = 12 cm
Calculating the new volume:
\[ V_{\text{new}} = \pi (18^2)(12) = \pi (324)(12) = 3888\pi \text{ cm}^3 \]
Comparing the New Volume with the Original Volume
We can also compare this new volume to the volume of the smaller cylinder:
\[ \text{New ratio} = \frac{V_{\text{new}}}{V_{\text{small}}} = \frac{3888\pi}{486\pi} \]
Again, the \(\pi\) cancels out:
\[ \text{New ratio} = \frac{3888}{486} = 8 \]
Thus, if both the radius and the height of the cylinder are doubled, the volume of the new cylinder becomes 8 times greater than the volume of the original smaller cylinder.
Summary
- The volume of the larger cylinder (radius doubled, height remains) is 4 times greater than the smaller cylinder.
- If both the height and radius are doubled, the volume is 8 times greater than the original smaller cylinder's volume.
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