Each of the radius of the base and the height of a right circular cylinder is increased by 10%. The volume of the cylinder is increased by

Options

A. `3.31%`
B. `14.5%`
C. `33.1%`
D. `19.5%`

Answer

To find the increase in the volume of a right circular cylinder when both the radius and height are increased by 10%, we can follow these steps:

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

### Step 2: Define the original dimensions
Let the original radius be \( r \) and the original height be \( h \).

### Step 3: Calculate the new dimensions after the increase
Since both the radius and height are increased by 10%, we can express the new dimensions as:
- New radius \( r' = r + 0.1r = 1.1r \)
- New height \( h' = h + 0.1h = 1.1h \)

### Step 4: Calculate the new volume
Now, we can calculate the new volume \( V' \) using the new dimensions:
\[ V' = \pi (r')^2 (h') = \pi (1.1r)^2 (1.1h) \]
Calculating \( (1.1r)^2 \):
\[ (1.1r)^2 = 1.21r^2 \]
Thus, the new volume becomes:
\[ V' = \pi (1.21r^2)(1.1h) = \pi (1.21 \times 1.1) r^2 h \]
Calculating \( 1.21 \times 1.1 \):
\[ 1.21 \times 1.1 = 1.331 \]
So, the new volume is:
\[ V' = \pi (1.331r^2 h) \]

### Step 5: Calculate the increase in volume
Now, we can find the increase in volume:
\[ \text{Increase in Volume} = V' - V = \pi (1.331r^2 h) - \pi (r^2 h) = \pi r^2 h (1.331 - 1) = \pi r^2 h (0.331) \]

### Step 6: Calculate the percentage increase in volume
To find the percentage increase, we use the formula:
\[ \text{Percentage Increase} = \left( \frac{\text{Increase in Volume}}{V} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage Increase} = \left( \frac{\pi r^2 h (0.331)}{\pi r^2 h} \right) \times 100 = 0.331 \times 100 = 33.1\% \]

### Final Answer
The volume of the cylinder is increased by **33.1%**.

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