If the radius of sphere is decreased by 10%, then by what per cent volume of sphere will decrease ?

Options

A. 30
B. `27.1`
C. `29.3`
D. `28.5`

Answer

To find the percentage decrease in the volume of a sphere when its radius is decreased by 10%, we can follow these steps:

### Step 1: Understand the formula for the volume of a sphere.
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

### Step 2: Calculate the original volume.
Let the original radius of the sphere be \( r \). Then the original volume \( V_1 \) is:
\[ V_1 = \frac{4}{3} \pi r^3 \]

### Step 3: Determine the new radius after a 10% decrease.
If the radius is decreased by 10%, the new radius \( r' \) will be:
\[ r' = r - 0.1r = 0.9r \]

### Step 4: Calculate the new volume with the decreased radius.
Now, we can calculate the new volume \( V_2 \) using the new radius:
\[ V_2 = \frac{4}{3} \pi (r')^3 = \frac{4}{3} \pi (0.9r)^3 \]
Calculating \( (0.9r)^3 \):
\[ (0.9r)^3 = 0.729r^3 \]
Thus, the new volume \( V_2 \) becomes:
\[ V_2 = \frac{4}{3} \pi (0.729r^3) = \frac{4}{3} \pi \cdot 0.729r^3 \]

### Step 5: Calculate the decrease in volume.
Now, we find the decrease in volume \( \Delta V \):
\[ \Delta V = V_1 - V_2 = \frac{4}{3} \pi r^3 - \frac{4}{3} \pi \cdot 0.729r^3 \]
Factoring out \( \frac{4}{3} \pi r^3 \):
\[ \Delta V = \frac{4}{3} \pi r^3 (1 - 0.729) = \frac{4}{3} \pi r^3 \cdot 0.271 \]

### Step 6: Calculate the percentage decrease in volume.
The percentage decrease in volume is given by:
\[ \text{Percentage Decrease} = \left(\frac{\Delta V}{V_1}\right) \times 100 \]
Substituting the values:
\[ \text{Percentage Decrease} = \left(\frac{\frac{4}{3} \pi r^3 \cdot 0.271}{\frac{4}{3} \pi r^3}\right) \times 100 = 0.271 \times 100 = 27.1\% \]

### Final Answer:
The volume of the sphere will decrease by **27.1%**.

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