The radius of a metal sphere at room temperature T is R, and the coefficient of linear expansion of the metal is `alpha`. The sphere is heated a little by a temperature `Delta T` so that its new temperature is `T+ Delta T`. The increase in the volume of the sphere is approximately

Options

A. `2pi R alpha Delta T`
B. `pi R^(2) alpha Delta T`
C. `4pi R^(3)alpha Delta T//3`
D. `4pi R^(3) alpha Delta T`

Explanation

To find the increase in the volume of a metal sphere when it is heated, we can follow these steps:

### Step 1: Understand the Given Information
- The radius of the sphere at room temperature \( T \) is \( R \).
- The coefficient of linear expansion of the metal is \( \alpha \).
- The increase in temperature is \( \Delta T \).

### Step 2: Relate Linear Expansion to Volume Expansion
The coefficient of volume expansion \( \gamma \) is related to the coefficient of linear expansion \( \alpha \) by the formula:
\[ \gamma = 3\alpha \]

### Step 3: Write the Volume of the Sphere
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi R^3 \]

### Step 4: Calculate the Change in Volume
The change in volume \( \Delta V \) due to the temperature change can be calculated using the formula:
\[ \Delta V = \gamma \cdot V \cdot \Delta T \]
Substituting \( \gamma = 3\alpha \) and \( V = \frac{4}{3} \pi R^3 \):
\[ \Delta V = 3\alpha \cdot \left(\frac{4}{3} \pi R^3\right) \cdot \Delta T \]

### Step 5: Simplify the Expression
Now, simplify the expression:
\[ \Delta V = 4\pi R^3 \alpha \Delta T \]

### Final Answer
Thus, the increase in the volume of the sphere when heated by \( \Delta T \) is approximately:
\[ \Delta V = 4\pi R^3 \alpha \Delta T \]

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