The radius of a sphere is doubled. What percent of its volume is increased ?

Question

Accepted Answer

To determine the percentage increase in the volume of a sphere when its radius is doubled, we need to understand the relationship between the radius and the volume of a sphere.

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

2. **Initial Volume**: Let's denote the initial radius as $ r $. The initial volume $ V_1 $ is:
   $$
   V_1 = \frac{4}{3} \pi r^3
   $$

3. **New Volume with Doubled Radius**: If the radius is doubled, the new radius becomes $ 2r $. The new volume $ V_2 $ is:
   $$
   V_2 = \frac{4}{3} \pi (2r)^3 = \frac{4}{3} \pi \times 8r^3 = \frac{32}{3} \pi r^3
   $$

4. **Increase in Volume**: The increase in volume $ \Delta V $ is:
   $$
   \Delta V = V_2 - V_1 = \frac{32}{3} \pi r^3 - \frac{4}{3} \pi r^3 = \frac{28}{3} \pi r^3
   $$

5. **Percentage Increase**: The percentage increase in volume is calculated as:
   $$
   \text{Percentage Increase} = \left( \frac{\Delta V}{V_1} \right) \times 100 = \left( \frac{\frac{28}{3} \pi r^3}{\frac{4}{3} \pi r^3} \right) \times 100
   $$
   Simplifying this expression:
   $$
   = \left( \frac{28}{4} \right) \times 100 = 7 \times 100 = 700\%
   $$

Therefore, the volume of the sphere increases by 700% when the radius is doubled, which justifies why Option 4 is correct.

**Explanation for Incorrect Options**:

- **Option 1 (200%)**: This option is incorrect because it underestimates the increase. Doubling the radius results in an eightfold increase in volume, not just a doubling.

- **Option 2 (450%)**: This option is incorrect because it does not accurately reflect the mathematical calculation of the volume increase, which is significantly higher.

- **Option 3 (550%)**: This option is incorrect because it also underestimates the actual increase in volume, which is calculated to be 700%.

In conclusion, the correct answer is Option 4, as the volume of the sphere increases by 700% when the radius is doubled.