If the diameter of sphere is decreased by 25%, then by what percent the curved surface area will be decreased?

Options

A. 34 3/4%
B. 75%
C. 25%
D. 43 3/4%

Answer

To solve the problem of how much the curved surface area (CSA) of a sphere decreases when its diameter is decreased by 25%, we will follow these steps:

### Step 1: Understand the relationship between diameter and radius
The radius (r) of a sphere is half of its diameter (d). If the diameter is decreased by 25%, we can express this mathematically.

### Step 2: Let the initial diameter be \( d \)
Assume the initial diameter of the sphere is \( d \). After a 25% decrease, the new diameter \( d' \) can be calculated as:
\[ d' = d - 0.25d = 0.75d \]

### Step 3: Calculate the initial and new radius
The initial radius \( r \) is:
\[ r = \frac{d}{2} \]
The new radius \( r' \) after the decrease in diameter is:
\[ r' = \frac{d'}{2} = \frac{0.75d}{2} = 0.375d \]

### Step 4: Calculate the initial and new curved surface area
The formula for the curved surface area (CSA) of a sphere is given by:
\[ \text{CSA} = 4\pi r^2 \]
The initial CSA \( \text{CSA}_1 \) is:
\[ \text{CSA}_1 = 4\pi \left(\frac{d}{2}\right)^2 = 4\pi \left(\frac{d^2}{4}\right) = \pi d^2 \]
The new CSA \( \text{CSA}_2 \) is:
\[ \text{CSA}_2 = 4\pi \left(0.375d\right)^2 = 4\pi \left(0.140625d^2\right) = 0.5625\pi d^2 \]

### Step 5: Calculate the decrease in curved surface area
The decrease in CSA is:
\[ \text{Decrease} = \text{CSA}_1 - \text{CSA}_2 = \pi d^2 - 0.5625\pi d^2 = (1 - 0.5625)\pi d^2 = 0.4375\pi d^2 \]

### Step 6: Calculate the percentage decrease in CSA
To find the percentage decrease, we use the formula:
\[ \text{Percentage Decrease} = \left(\frac{\text{Decrease}}{\text{CSA}_1}\right) \times 100 \]
Substituting the values:
\[ \text{Percentage Decrease} = \left(\frac{0.4375\pi d^2}{\pi d^2}\right) \times 100 = 0.4375 \times 100 = 43.75\% \]

### Final Answer
The curved surface area of the sphere will decrease by **43.75%** when the diameter is decreased by 25%.

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