If the radius of a cylinder is decreased by 8%, while its height is increased by 4%, what will be the effect on volume?

Options

A. 11.9744% (decrease)
B. 11.9744% (increase)
C. 12.4678% (decrease)
D. 12.4678% (increase)

Answer

To determine the effect on the volume of a cylinder when its radius is decreased by 8% and its height is increased by 4%, we can follow these steps:

### Step 1: Understand the formula for volume
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: Identify the changes in dimensions
- The radius is decreased by 8%, which means the new radius \( r' \) can be expressed as:
\[ r' = r \times (1 - 0.08) = r \times 0.92 \]
- The height is increased by 4%, which means the new height \( h' \) can be expressed as:
\[ h' = h \times (1 + 0.04) = h \times 1.04 \]

### Step 3: Substitute the new dimensions into the volume formula
Now, substituting \( r' \) and \( h' \) into the volume formula, we get:
\[ V' = \pi (r')^2 (h') = \pi (0.92r)^2 (1.04h) \]

### Step 4: Simplify the expression for the new volume
Expanding the expression:
\[ V' = \pi (0.92^2 r^2) (1.04h) = \pi (0.8464 r^2) (1.04h) \]
\[ V' = \pi (0.8464 \times 1.04) r^2 h \]
Calculating \( 0.8464 \times 1.04 \):
\[ 0.8464 \times 1.04 = 0.881856 \]
Thus, we have:
\[ V' = \pi (0.881856 r^2 h) \]

### Step 5: Calculate the percentage change in volume
To find the percentage change in volume, we compare the new volume \( V' \) with the original volume \( V \):
\[ \text{Percentage change} = \left( \frac{V' - V}{V} \right) \times 100 \]
Substituting \( V' \) and \( V \):
\[ \text{Percentage change} = \left( \frac{0.881856 V - V}{V} \right) \times 100 = (0.881856 - 1) \times 100 \]
\[ = -0.118144 \times 100 = -11.8144\% \]

### Conclusion
The volume of the cylinder decreases by approximately **11.81%**.

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