If a solid right circular cylinder made of iron is heated to increase its radius and height by `1%` each, then by how much per cent is the volume of the solid increased ?

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 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 and \( h \) is the height of the cylinder.

### Step 2: Determine the new dimensions after the increase
If the radius and height are increased by 1%, the new radius \( r' \) and new height \( h' \) can be calculated as follows:
\[ r' = r + 0.01r = 1.01r \]
\[ h' = h + 0.01h = 1.01h \]

### Step 3: Calculate the new volume with the increased dimensions
Substituting the new dimensions into the volume formula gives:
\[ V' = \pi (r')^2 (h') = \pi (1.01r)^2 (1.01h) \]
Calculating \( (1.01r)^2 \):
\[ (1.01r)^2 = 1.0201r^2 \]
Thus, the new volume becomes:
\[ V' = \pi (1.0201r^2)(1.01h) = \pi (1.0201)(1.01)(r^2)(h) \]
Now, calculating \( 1.0201 \times 1.01 \):
\[ 1.0201 \times 1.01 = 1.030301 \]
So, the new volume is:
\[ V' = \pi r^2 h \times 1.030301 \]

### Step 4: Calculate the increase in volume
The increase in volume \( \Delta V \) is:
\[ \Delta V = V' - V = \pi r^2 h \times 1.030301 - \pi r^2 h \]
Factoring out \( \pi r^2 h \):
\[ \Delta V = \pi r^2 h (1.030301 - 1) = \pi r^2 h \times 0.030301 \]

### Step 5: Calculate the percentage increase in volume
The percentage increase in volume is given by:
\[ \text{Percentage Increase} = \left( \frac{\Delta V}{V} \right) \times 100 = \left( \frac{\pi r^2 h \times 0.030301}{\pi r^2 h} \right) \times 100 \]
This simplifies to:
\[ \text{Percentage Increase} = 0.030301 \times 100 = 3.0301\% \]

### Final Answer
The volume of the solid is increased by approximately **3.03%**.

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