The diameter of a sphere is increased by 25%. The percent increase in its volume is :

Question

Accepted Answer

To determine the percent increase in the volume of a sphere when its diameter is increased by 25%, we need to understand the relationship between the diameter and the volume of a sphere.

1. **Volume of a Sphere Formula**:  
   The volume $ V $ of a sphere is given by the formula:
   $$
   V = \frac{4}{3} \pi r^3
   $$
   where $ r $ is the radius of the sphere.

2. **Initial and New Radius**:  
   If the original diameter is $ d $, the original radius $ r $ is $ \frac{d}{2} $.  
   When the diameter is increased by 25%, the new diameter becomes $ 1.25d $, and the new radius $ r' $ is $ \frac{1.25d}{2} = 1.25 \times \frac{d}{2} = 1.25r $.

3. **New Volume Calculation**:  
   The new volume $ V' $ with the increased radius $ r' = 1.25r $ is:
   $$
   V' = \frac{4}{3} \pi (1.25r)^3 = \frac{4}{3} \pi (1.953125r^3)
   $$
   Simplifying, we get:
   $$
   V' = 1.953125 \times \frac{4}{3} \pi r^3
   $$

4. **Percent Increase in Volume**:  
   The percent increase in volume is calculated as:
   $$
   \text{Percent Increase} = \left(\frac{V' - V}{V}\right) \times 100\%
   $$
   Substituting the values, we have:
   $$
   \text{Percent Increase} = \left(\frac{1.953125V - V}{V}\right) \times 100\% = (1.953125 - 1) \times 100\% = 0.953125 \times 100\% = 95.3125\%
   $$

5. **Rounding**:  
   The calculated percent increase is 95.3125%, which rounds to 95.31%.

Therefore, the correct answer is Option 4: 95.31%.

**Explanation for Incorrect Options**:

- **Option 1 (59.13%)**: This value is significantly lower than the calculated increase of 95.31%. It does not account for the cubic relationship between the radius and the volume.
  
- **Option 2 (59.31%)**: Similar to Option 1, this value is incorrect as it underestimates the increase in volume due to the cubic nature of the volume formula.
  
- **Option 3 (95.13%)**: Although close, this option is slightly off due to rounding. The precise calculation yields 95.31%, making Option 4 the correct choice.

In conclusion, the correct answer is Option 4: 95.31%, as it accurately reflects the percent increase in volume when the diameter of a sphere is increased by 25%.