If a solid right circular cylinder is made of iron is heated to increae its radius and height by 1% each , then the volume of the solid is increased by :

Options

A. `1.01%`
B. `3.03%`
C. `2.02%`
D. `1.2%`

Answer

To find the percentage increase in the volume of a solid right circular cylinder when its radius and height are both increased by 1%, we can follow these steps:

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

### Step 2: Set Initial Values
Let's assume the initial radius \( r \) and height \( h \) of the cylinder are both 100 units. Thus:
\[ r = 100, \quad h = 100 \]

### Step 3: Calculate Initial Volume
Using the initial values, the initial volume \( V_1 \) can be calculated as:
\[ V_1 = \pi (100)^2 (100) = \pi \times 10000 \times 100 = 1000000\pi \]

### Step 4: Calculate New Dimensions After Increase
Since both the radius and height are increased by 1%, the new radius \( r' \) and new height \( h' \) will be:
\[ r' = r + 0.01r = 100 + 1 = 101 \]
\[ h' = h + 0.01h = 100 + 1 = 101 \]

### Step 5: Calculate New Volume
Now, we can calculate the new volume \( V_2 \) with the new dimensions:
\[ V_2 = \pi (101)^2 (101) \]
Calculating \( (101)^2 \):
\[ (101)^2 = 10201 \]
Thus, the new volume becomes:
\[ V_2 = \pi \times 10201 \times 101 = 1030301\pi \]

### Step 6: Calculate the Increase in Volume
The increase in volume \( \Delta V \) is given by:
\[ \Delta V = V_2 - V_1 = 1030301\pi - 1000000\pi = (1030301 - 1000000)\pi = 30301\pi \]

### Step 7: Calculate Percentage Increase
To find the percentage increase in volume, we use the formula:
\[ \text{Percentage Increase} = \left(\frac{\Delta V}{V_1}\right) \times 100 \]
Substituting the values we found:
\[ \text{Percentage Increase} = \left(\frac{30301\pi}{1000000\pi}\right) \times 100 = \left(\frac{30301}{1000000}\right) \times 100 \]
This simplifies to:
\[ \text{Percentage Increase} = \frac{30301}{10000} = 3.0301\% \]

### Conclusion
Thus, the volume of the solid is increased by approximately **3.03%**.

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