On increasing each of the radius of the base and the height of a cone by 20% its volume will be increased by

Options

A. `52%`
B. `75%`
C. `62%`
D. `72%`

Explanation

To solve the problem of how much the volume of a cone increases when both the radius and height are increased by 20%, we can follow these steps:

### Step 1: Understand the formula for the volume of a cone
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base and \( h \) is the height.

### Step 2: Determine the new dimensions after the increase
If we increase the radius \( r \) and height \( h \) by 20%, the new radius \( r' \) and new height \( h' \) can be calculated as follows:
\[ r' = r + 0.2r = 1.2r = \frac{6}{5}r \]
\[ h' = h + 0.2h = 1.2h = \frac{6}{5}h \]

### Step 3: Calculate the new volume with the increased dimensions
Now, substituting the new radius and height into the volume formula, we get the new volume \( V' \):
\[ V' = \frac{1}{3} \pi (r')^2 (h') = \frac{1}{3} \pi \left(\frac{6}{5}r\right)^2 \left(\frac{6}{5}h\right) \]
Calculating \( (r')^2 \):
\[ (r')^2 = \left(\frac{6}{5}r\right)^2 = \frac{36}{25}r^2 \]
Thus, the new volume becomes:
\[ V' = \frac{1}{3} \pi \left(\frac{36}{25}r^2\right) \left(\frac{6}{5}h\right) = \frac{1}{3} \pi \frac{216}{125} r^2 h \]

### Step 4: Relate the new volume to the old volume
We can express the new volume \( V' \) in terms of the old volume \( V \):
\[ V' = \frac{216}{125} \cdot \frac{1}{3} \pi r^2 h = \frac{216}{125} V \]

### Step 5: Calculate the increase in volume
To find the increase in volume, we subtract the old volume from the new volume:
\[ \text{Increase in Volume} = V' - V = \frac{216}{125} V - V = \left(\frac{216}{125} - 1\right)V = \left(\frac{216 - 125}{125}\right)V = \frac{91}{125}V \]

### Step 6: Calculate the percentage increase
To find the percentage increase in volume, we use the formula:
\[ \text{Percentage Increase} = \left(\frac{\text{Increase in Volume}}{V}\right) \times 100 = \left(\frac{\frac{91}{125}V}{V}\right) \times 100 = \frac{91}{125} \times 100 \]
Calculating this gives:
\[ \frac{91 \times 100}{125} = 72.8\% \]

### Final Answer
Thus, the volume of the cone will increase by approximately **72.8%** when both the radius and height are increased by 20%.

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