If we reduce the height of a cyliner by `1/4` and double the radius , what will be the impact on the volume ?

Answer

To determine the impact on the volume of a cylinder when the height is reduced by \( \frac{1}{4} \) and the radius is doubled, we can follow these steps:

### Step 1: Write the formula for the volume of a cylinder.
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

### Step 2: Define the original dimensions of the cylinder.
Let the original radius be \( r \) and the original height be \( h \). Therefore, the original volume \( V_{\text{original}} \) is:
\[ V_{\text{original}} = \pi r^2 h \]

### Step 3: Determine the new dimensions after changes.
- The height is reduced by \( \frac{1}{4} \), so the new height \( h_{\text{new}} \) is:
\[ h_{\text{new}} = h - \frac{1}{4}h = \frac{3}{4}h \]
- The radius is doubled, so the new radius \( r_{\text{new}} \) is:
\[ r_{\text{new}} = 2r \]

### Step 4: Calculate the new volume with the new dimensions.
Now, substituting the new radius and height into the volume formula, the new volume \( V_{\text{new}} \) is:
\[ V_{\text{new}} = \pi (r_{\text{new}})^2 (h_{\text{new}}) = \pi (2r)^2 \left(\frac{3}{4}h\right) \]
Calculating this gives:
\[ V_{\text{new}} = \pi (4r^2) \left(\frac{3}{4}h\right) = \pi \cdot 4r^2 \cdot \frac{3}{4}h = 3\pi r^2 h \]

### Step 5: Compare the new volume with the original volume.
Now, we can compare the new volume \( V_{\text{new}} \) with the original volume \( V_{\text{original}} \):
- Original volume:
\[ V_{\text{original}} = \pi r^2 h \]
- New volume:
\[ V_{\text{new}} = 3\pi r^2 h \]

### Step 6: Determine the impact on the volume.
To find the impact on the volume, we can express the new volume in terms of the original volume:
\[ V_{\text{new}} = 3 \times V_{\text{original}} \]

### Conclusion:
Thus, the new volume is three times the original volume. Therefore, the impact on the volume is that it increases by a factor of 3.

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