An ideal gas is initially at temperature T and volume V. ITS volume is increased by `DeltaV` due to an increase in temperature `DeltaT`, pressure remaining constant. The quantity `delta = DeltaV//V DeltaT` varies with temperature as

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Explanation

To solve the problem, we will analyze the behavior of an ideal gas when its volume increases due to a change in temperature while keeping the pressure constant. We will derive the relationship between the given quantities step by step.

### Step-by-Step Solution:

1. **Understand the Ideal Gas Law**:
The ideal gas law is given by the equation:
\[ PV = nRT \]
where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature.

2. **Initial Conditions**:
Initially, the gas is at temperature \( T \) and volume \( V \). When the temperature increases by \( \Delta T \), the volume increases by \( \Delta V \) while the pressure remains constant.

3. **Express Volume in Terms of Temperature**:
Since pressure is constant, we can express the volume as:
\[ V = \frac{nR}{P} T \]

4. **Change in Volume**:
When the temperature increases by \( \Delta T \), the new volume \( V' \) can be expressed as:
\[ V' = \frac{nR}{P} (T + \Delta T) = \frac{nR}{P} T + \frac{nR}{P} \Delta T \]

5. **Calculate the Change in Volume**:
The change in volume \( \Delta V \) is given by:
\[ \Delta V = V' - V = \frac{nR}{P} \Delta T \]

6. **Formulate the Quantity \( \delta \)**:
We need to find the quantity:
\[ \delta = \frac{\Delta V}{V \Delta T} \]
Substituting \( \Delta V \) and \( V \) into this equation:
\[ \delta = \frac{\frac{nR}{P} \Delta T}{\frac{nR}{P} T \Delta T} \]
Simplifying this gives:
\[ \delta = \frac{1}{T} \]

7. **Analyze the Variation of \( \delta \) with Temperature**:
The expression \( \delta = \frac{1}{T} \) indicates that \( \delta \) varies inversely with temperature \( T \).

8. **Graphical Representation**:
As \( T \) approaches 0, \( \delta \) approaches infinity, and as \( T \) approaches infinity, \( \delta \) approaches 0. This relationship is characteristic of a rectangular hyperbola.

### Conclusion:
The quantity \( \delta \) varies with temperature as:
\[ \delta \propto \frac{1}{T} \]