If radius of a cylinder is decreased by 4% , while its height is increased by 2%, then what will be effect on volume ?

Answer

To find the effect on the volume of a cylinder when the radius is decreased by 4% and the height is increased by 2%, we can follow these steps:

### Step 1: Understand the Volume Formula
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.

**Hint:** Remember that the volume depends on both the radius and height of the cylinder.

### Step 2: Calculate the New Radius
If the radius is decreased by 4%, the new radius \( r' \) can be calculated as:
\[ r' = r - 0.04r = 0.96r \]

**Hint:** To find the new radius after a percentage decrease, multiply the original radius by \( (1 - \text{percentage decrease}) \).

### Step 3: Calculate the New Height
If the height is increased by 2%, the new height \( h' \) can be calculated as:
\[ h' = h + 0.02h = 1.02h \]

**Hint:** To find the new height after a percentage increase, multiply the original height by \( (1 + \text{percentage increase}) \).

### Step 4: Substitute the New Values into the Volume Formula
Now substitute the new radius and height into the volume formula:
\[ V' = \pi (r')^2 (h') = \pi (0.96r)^2 (1.02h) \]

### Step 5: Simplify the Expression
Now simplify the expression for the new volume:
\[ V' = \pi (0.9216r^2)(1.02h) = \pi (0.9216 \cdot 1.02) r^2 h \]
Calculating \( 0.9216 \cdot 1.02 \):
\[ 0.9216 \cdot 1.02 = 0.940992 \]
Thus, we have:
\[ V' = \pi (0.940992) r^2 h \]

### Step 6: Compare the New Volume with the Original Volume
The original volume \( V \) is:
\[ V = \pi r^2 h \]
Now, we can express the new volume in terms of the original volume:
\[ V' = 0.940992 V \]

### Step 7: Determine the Effect on Volume
The new volume \( V' \) is approximately 94.1% of the original volume \( V \). This indicates that the volume has decreased.

**Final Conclusion:** The volume of the cylinder decreases by approximately 5.9%.