If the radius of a cylinder is decreased by 40% and the height is increased by 60% to form a new cylinder, then the volume will be decreased by:

Options

A. `41.5% `
B. `32.4% `
C. `40.5% `
D. `42.4% `

Answer

To solve the problem, we need to calculate the volume of the original cylinder and the volume of the new cylinder after the changes in radius and height.

### Step 1: Understand the formula for the volume of a cylinder
The volume \( V \) of a 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 original dimensions
Let the original radius of the cylinder be \( r \) and the original height be \( h \). Therefore, the volume of the original cylinder is:
\[ V_{original} = \pi r^2 h \]

### Step 3: Calculate the new radius
The radius is decreased by 40%. This means the new radius \( r' \) is:
\[ r' = r - 0.4r = 0.6r \]

### Step 4: Calculate the new height
The height is increased by 60%. This means the new height \( h' \) is:
\[ h' = h + 0.6h = 1.6h \]

### Step 5: Calculate the volume of the new cylinder
Now, we can calculate the volume of the new cylinder using the new dimensions:
\[ V_{new} = \pi (r')^2 (h') = \pi (0.6r)^2 (1.6h) \]
Calculating \( (0.6r)^2 \):
\[ (0.6r)^2 = 0.36r^2 \]
Thus, the volume of the new cylinder becomes:
\[ V_{new} = \pi (0.36r^2)(1.6h) = \pi (0.576r^2h) \]

### Step 6: Calculate the percentage decrease in volume
Now we can find the percentage decrease in volume. The original volume is \( V_{original} = \pi r^2 h \) and the new volume is \( V_{new} = \pi (0.576r^2h) \).

The decrease in volume is:
\[ \text{Decrease} = V_{original} - V_{new} = \pi r^2 h - \pi (0.576r^2h) = \pi r^2 h (1 - 0.576) = \pi r^2 h (0.424) \]

To find the percentage decrease:
\[ \text{Percentage Decrease} = \left(\frac{\text{Decrease}}{V_{original}}\right) \times 100 = \left(\frac{0.424 \pi r^2 h}{\pi r^2 h}\right) \times 100 = 42.4\% \]

### Final Answer
The volume of the cylinder will be decreased by approximately **42.4%**.