Solution:

The temperature gradient is the rate at which temperature changes with respect to distance. It's given as \( 80^\circ \text{C/m} \), and the length of the rod is \( 0.5 \, \text{m} \).

To find the temperature difference across the rod, we use the formula:

\[
\Delta T = \text{Temperature gradient} \times \text{Length}
\]

Substitute the values:

\[
\Delta T = 80^\circ \text{C/m} \times 0.5 \, \text{m} = 40^\circ \text{C}
\]

Now, the temperature at the hotter end is given as \( 30^\circ \text{C} \), so the temperature at the colder end is:

\[
T_{\text{colder end}} = T_{\text{hotter end}} - \Delta T
\]

Substituting the values:

\[
T_{\text{colder end}} = 30^\circ \text{C} - 40^\circ \text{C} = -10^\circ \text{C}
\]

Therefore, the temperature at the colder end of the rod is \(-10^\circ \text{C}\).