To determine how many times greater the volume of the larger cylinder is compared to the smaller cylinder when the radius is doubled, we first need to calculate the volume 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.
Volume of the Smaller Cylinder:
- Radius \( r = 7 \) m
- Height \( h = 12 \) m
Substituting these values into the formula:
\[ V_{\text{small}} = \pi (7^2)(12) = \pi (49)(12) = 588\pi \text{ m}^3 \]
Volume of the Larger Cylinder (with doubled radius):
- New Radius \( r = 2 \times 7 = 14 \) m
- Height remains the same \( h = 12 \) m
Now substituting the new radius into the volume formula:
\[ V_{\text{large}} = \pi (14^2)(12) = \pi (196)(12) = 2352\pi \text{ m}^3 \]
Comparison of Volumes:
Now we find out how many times greater the volume of the larger cylinder is compared to the smaller cylinder:
\[ \text{Factor of increase} = \frac{V_{\text{large}}}{V_{\text{small}}} = \frac{2352\pi}{588\pi} \]
The \(\pi\) cancels out:
\[ \text{Factor of increase} = \frac{2352}{588} \]
Now, divide:
\[ \frac{2352 \div 588} = 4 \]
Thus, the volume of the larger cylinder is 4 times greater than the volume of the smaller cylinder when the radius is doubled.
If the Height Were Also Doubled:
If both the radius and the height were doubled, we will look at the new dimensions:
- New Radius \( r = 14 \) m (double of 7 m)
- New Height \( h = 24 \) m (double of 12 m)
Calculating the volume of the new cylinder with doubled radius and height:
\[ V_{\text{new}} = \pi (14^2)(24) = \pi (196)(24) = 4704\pi \text{ m}^3 \]
We can compare this new volume \( V_{\text{new}} \) to the original volume of the small cylinder \( V_{\text{small}} \):
\[ \text{Factor of increase from small to new} = \frac{V_{\text{new}}}{V_{\text{small}}} = \frac{4704\pi}{588\pi} \]
Again, the \(\pi\) cancels out:
\[ \text{Factor of increase} = \frac{4704}{588} = 8 \]
Thus, if both the radius and height of the cylinder are doubled, the volume of the new cylinder is 8 times greater than that of the original smaller cylinder.