An aluminium sphere of `20cm` diameter is heated from `0^(@)C` to `100^(@)C`. Its volume changes by (given that the coefficient of linear expanison for aluminium `(alpha_(Al) = 23 xx 10^(-6//0)C)`

Options

A. `2.89 c c`
B. `9.28 c c`
C. `49.8 c c`
D. `28.9 c c`

Explanation

To solve the problem of determining the volume change of an aluminum sphere when heated, we can follow these steps:

### Step 1: Determine the radius of the sphere
The diameter of the sphere is given as 20 cm. The radius \( r \) can be calculated as:
\[ r = \frac{\text{diameter}}{2} = \frac{20 \, \text{cm}}{2} = 10 \, \text{cm} = 0.1 \, \text{m} \]

### Step 2: Calculate the initial volume of the sphere
The volume \( V_0 \) of a sphere is given by the formula:
\[ V_0 = \frac{4}{3} \pi r^3 \]
Substituting the value of \( r \):
\[ V_0 = \frac{4}{3} \pi (0.1)^3 = \frac{4}{3} \pi (0.001) = \frac{4\pi}{3000} \approx 0.00418879 \, \text{m}^3 \approx 418.88 \, \text{cm}^3 \]

### Step 3: Determine the change in temperature
The initial temperature is \( 0^\circ C \) and the final temperature is \( 100^\circ C \). Therefore, the change in temperature \( \Delta T \) is:
\[ \Delta T = 100^\circ C - 0^\circ C = 100^\circ C \]

### Step 4: Calculate the coefficient of volume expansion
The coefficient of volume expansion \( \gamma \) is related to the coefficient of linear expansion \( \alpha \) by the formula:
\[ \gamma = 3\alpha \]
Given \( \alpha = 23 \times 10^{-6} \, \text{°C}^{-1} \):
\[ \gamma = 3 \times (23 \times 10^{-6}) = 69 \times 10^{-6} \, \text{°C}^{-1} \]

### Step 5: Calculate the change in volume
The change in volume \( \Delta V \) can be calculated using the formula:
\[ \Delta V = V_0 \cdot \gamma \cdot \Delta T \]
Substituting the values:
\[ \Delta V = 418.88 \, \text{cm}^3 \cdot (69 \times 10^{-6}) \cdot 100 \]
Calculating this gives:
\[ \Delta V = 418.88 \cdot 0.000069 \cdot 100 \approx 28.9 \, \text{cm}^3 \]

### Final Answer
The change in volume of the aluminum sphere when heated from \( 0^\circ C \) to \( 100^\circ C \) is approximately:
\[ \Delta V \approx 28.9 \, \text{cm}^3 \]

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