If radius of a sphere is decreased by 24%, by what per cent does its surface area decrease?

Options

A. `44%`
B. `49%`
C. `42.24%`
D. `46.2%`

Answer

To solve the problem of how much the surface area of a sphere decreases when its radius is decreased by 24%, we can follow these steps:

### Step 1: Understand the relationship between radius and surface area
The surface area \( S \) of a sphere is given by the formula:
\[ S = 4\pi r^2 \]
where \( r \) is the radius of the sphere.

### Step 2: Calculate the new radius after a 24% decrease
If the radius is decreased by 24%, the new radius \( r' \) can be calculated as:
\[ r' = r - 0.24r = 0.76r \]

### Step 3: Calculate the new surface area
Substituting the new radius into the surface area formula:
\[ S' = 4\pi (r')^2 = 4\pi (0.76r)^2 \]
Calculating \( (0.76r)^2 \):
\[ (0.76r)^2 = 0.5776r^2 \]
Thus, the new surface area becomes:
\[ S' = 4\pi (0.5776r^2) = 4\pi \cdot 0.5776 \cdot r^2 = 2.3104\pi r^2 \]

### Step 4: Calculate the percentage decrease in surface area
The original surface area \( S \) is:
\[ S = 4\pi r^2 \]
The decrease in surface area \( \Delta S \) is:
\[ \Delta S = S - S' = 4\pi r^2 - 2.3104\pi r^2 = (4 - 2.3104)\pi r^2 = 1.6896\pi r^2 \]
To find the percentage decrease:
\[ \text{Percentage decrease} = \left(\frac{\Delta S}{S}\right) \times 100 = \left(\frac{1.6896\pi r^2}{4\pi r^2}\right) \times 100 = \left(\frac{1.6896}{4}\right) \times 100 \]
Calculating this gives:
\[ \text{Percentage decrease} = 42.24\% \]

### Final Answer
The surface area decreases by **42.24%**.

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