Let A and B be two solid spheres such that the surface area of B is 300% more than the surface area of A. The volume of A is r% lower than the volume of B. Then value of r is equal to :

Question

Accepted Answer

To solve this problem, we need to understand the relationship between the surface area and volume of spheres. Let's break down the problem step by step:

1. **Surface Area Relationship:**
   - Let the surface area of sphere A be $ S_A $.
   - The surface area of sphere B is 300% more than that of sphere A. This means:
     $$
     S_B = S_A + 3 \times S_A = 4 \times S_A
     $$
   - The formula for the surface area of a sphere is $ 4\pi r^2 $. If the radius of sphere A is $ r_A $ and the radius of sphere B is $ r_B $, then:
     $$
     4\pi r_B^2 = 4 \times 4\pi r_A^2
     $$
     $$
     r_B^2 = 4 \times r_A^2
     $$
     $$
     r_B = 2 \times r_A
     $$

2. **Volume Relationship:**
   - The formula for the volume of a sphere is $ \frac{4}{3}\pi r^3 $.
   - The volume of sphere A is $ V_A = \frac{4}{3}\pi r_A^3 $.
   - The volume of sphere B is $ V_B = \frac{4}{3}\pi r_B^3 = \frac{4}{3}\pi (2r_A)^3 = \frac{4}{3}\pi \times 8r_A^3 = 8 \times \frac{4}{3}\pi r_A^3 = 8 \times V_A $.

3. **Percentage Decrease in Volume:**
   - We are told that the volume of A is $ r\% $ lower than the volume of B. This means:
     $$
     V_A = V_B - \frac{r}{100} \times V_B
     $$
     $$
     V_A = (1 - \frac{r}{100}) \times V_B
     $$
   - Since $ V_A = \frac{1}{8} \times V_B $, we have:
     $$
     \frac{1}{8} \times V_B = (1 - \frac{r}{100}) \times V_B
     $$
     $$
     \frac{1}{8} = 1 - \frac{r}{100}
     $$
     $$
     \frac{r}{100} = 1 - \frac{1}{8}
     $$
     $$
     \frac{r}{100} = \frac{7}{8}
     $$
     $$
     r = \frac{7}{8} \times 100 = 87.5
     $$

Thus, the value of $ r $ is 87.5, which corresponds to Option 2.

**Explanation for Incorrect Options:**

- **Option 1 (85.5):** This value does not satisfy the derived equation $ \frac{r}{100} = \frac{7}{8} $. The calculation shows that the correct percentage decrease is 87.5%, not 85.5%.
  
- **Option 3 (90.5):** Similarly, this value is incorrect because it exceeds the calculated percentage decrease of 87.5%. The mathematical derivation clearly shows that 87.5% is the correct value.
  
- **Option 4 (92.5):** This option is also incorrect as it is greater than the calculated value of 87.5%. The problem's conditions and mathematical relationships do not support this percentage.

Therefore, Option 2 is the correct answer, as it accurately reflects the percentage decrease in volume based on the given conditions.