If the radius of a right circular cylinder is decreased by 50% and its height is increased by 60%, its volume will be decreased by

Options

A. `10%`
B. `60%`
C. `40%`
D. `20%`

Answer

To solve the problem, we need to calculate the volume of a right circular cylinder before and after the changes in radius and height, and then determine the percentage decrease in volume.

### 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.

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

### Step 3: Calculate the new dimensions
- The radius is decreased by 50%. Therefore, the new radius \( r' \) is:
\[ r' = r - 0.5r = 0.5r \]
- The height is increased by 60%. Therefore, the new height \( h' \) is:
\[ h' = h + 0.6h = 1.6h \]

### Step 4: Calculate the initial volume
The initial volume \( V \) is:
\[ V = \pi r^2 h \]

### Step 5: Calculate the new volume
The new volume \( V' \) with the new dimensions is:
\[ V' = \pi (r')^2 (h') = \pi (0.5r)^2 (1.6h) \]
Calculating \( (0.5r)^2 \):
\[ (0.5r)^2 = 0.25r^2 \]
So, the new volume becomes:
\[ V' = \pi (0.25r^2)(1.6h) = 0.4\pi r^2 h \]

### Step 6: Calculate the decrease in volume
Now, we find the decrease in volume:
\[ \text{Decrease in volume} = V - V' = \pi r^2 h - 0.4\pi r^2 h = (1 - 0.4)\pi r^2 h = 0.6\pi r^2 h \]

### Step 7: Calculate the percentage decrease in volume
The percentage decrease in volume is given by:
\[ \text{Percentage decrease} = \left(\frac{\text{Decrease in volume}}{\text{Initial volume}}\right) \times 100 \]
Substituting the values:
\[ \text{Percentage decrease} = \left(\frac{0.6\pi r^2 h}{\pi r^2 h}\right) \times 100 = 0.6 \times 100 = 60\% \]

### Final Answer
The volume of the cylinder will be decreased by **60%**.

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