The radius of a sphere is 10cm. If the radius is increased by 1cm, then prove that volume of the sphere is increased by 33.1%.

Answer

To solve the problem step by step, we will calculate the volume of the sphere before and after the increase in radius and then find the percentage increase in volume.

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

Given that the original radius \( r = 10 \) cm, we can substitute this value into the formula:
\[ V_1 = \frac{4}{3} \pi (10)^3 \]
Calculating \( (10)^3 \):
\[ (10)^3 = 1000 \]
Now substituting back:
\[ V_1 = \frac{4}{3} \pi \times 1000 = \frac{4000}{3} \pi \, \text{cm}^3 \]

### Step 2: Calculate the new volume of the sphere after increasing the radius
The new radius after the increase is:
\[ R = 10 + 1 = 11 \, \text{cm} \]
Now, we calculate the new volume \( V_2 \) using the new radius:
\[ V_2 = \frac{4}{3} \pi (11)^3 \]
Calculating \( (11)^3 \):
\[ (11)^3 = 1331 \]
Now substituting back:
\[ V_2 = \frac{4}{3} \pi \times 1331 = \frac{5324}{3} \pi \, \text{cm}^3 \]

### Step 3: Calculate the increase in volume
The increase in volume \( \Delta V \) is given by:
\[ \Delta V = V_2 - V_1 \]
Substituting the values we calculated:
\[ \Delta V = \frac{5324}{3} \pi - \frac{4000}{3} \pi \]
Combining the fractions:
\[ \Delta V = \frac{5324 - 4000}{3} \pi = \frac{1324}{3} \pi \, \text{cm}^3 \]

### Step 4: Calculate the percentage increase in volume
The percentage increase in volume is given by:
\[ \text{Percentage Increase} = \left( \frac{\Delta V}{V_1} \right) \times 100\% \]
Substituting the values:
\[ \text{Percentage Increase} = \left( \frac{\frac{1324}{3} \pi}{\frac{4000}{3} \pi} \right) \times 100\% \]
The \( \frac{3 \pi}{3 \pi} \) cancels out:
\[ \text{Percentage Increase} = \left( \frac{1324}{4000} \right) \times 100\% \]
Calculating \( \frac{1324}{4000} \):
\[ \frac{1324}{4000} = 0.331 \]
Now multiplying by 100:
\[ \text{Percentage Increase} = 0.331 \times 100 = 33.1\% \]

### Conclusion
Thus, we have proved that the volume of the sphere increases by 33.1% when the radius is increased from 10 cm to 11 cm.

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