If the radius of a sphere is increased by 50%, then how much percentage increases in its lateral area?

Question

Accepted Answer

Here's a detailed explanation for the given question:

**Understanding the Problem:**
We need to find the percentage increase in the lateral area (which is the surface area for a sphere) when its radius is increased by 50%.

**Step-by-step Derivation:**

1.  **Formula for Lateral Area of a Sphere:**
    The lateral area (or surface area) of a sphere is given by the formula:
    `A = 4πr²`
    where `A` is the area and `r` is the radius.

2.  **Original Radius and Area:**
    Let the original radius be `r`.
    The original lateral area will be `A_original = 4πr²`.

3.  **New Radius:**
    The radius is increased by 50%.
    New radius `r_new = r + 50% of r`
    `r_new = r + 0.50r`
    `r_new = 1.5r`

4.  **New Lateral Area:**
    Now, calculate the new lateral area using the new radius `r_new`:
    `A_new = 4π(r_new)²`
    `A_new = 4π(1.5r)²`
    `A_new = 4π(2.25r²)`
    `A_new = 2.25 * (4πr²)`

Since `A_original = 4πr²`, we can substitute this into the equation for `A_new`:
    `A_new = 2.25 * A_original`

5.  **Percentage Increase in Area:**
    The percentage increase is calculated as:
    `Percentage Increase = ((A_new - A_original) / A_original) * 100%`
    `Percentage Increase = ((2.25 * A_original - A_original) / A_original) * 100%`
    `Percentage Increase = (1.25 * A_original / A_original) * 100%`
    `Percentage Increase = 1.25 * 100%`
    `Percentage Increase = 125%`

**Justification for Option 1 (125%):**
As calculated above, when the radius of a sphere is increased by 50%, the new radius becomes 1.5 times the original radius. Since the area is proportional to the square of the radius (`r²`), the new area becomes `(1.5)² = 2.25` times the original area. This means the area has increased by `2.25 - 1 = 1.25` times the original area. Converting this to a percentage, `1.25 * 100% = 125%`. Therefore, Option 1 is correct.

**Explanation for why other options are incorrect:**

*   **Option 2 (150%):** This would imply that the new area is 2.5 times the original area (`1 + 1.5 = 2.5`). Our calculation shows the new area is 2.25 times the original area, and the increase is 1.25 times the original area, not 1.5 times.
*   **Option 3 (200%):** This would imply that the new area is 3 times the original area (`1 + 2 = 3`). Our calculation shows the new area is 2.25 times the original area, and the increase is 1.25 times the original area, not 2 times.
*   **Option 4 (100%):** This would imply that the new area is 2 times the original area (`1 + 1 = 2`), meaning the area has doubled. Our calculation shows the new area is 2.25 times the original area, and the increase is 1.25 times the original area, not 1 time.

The final answer is $\boxed{	ext{125%}}$