Solution:

Using Newton's law of cooling:

Given:
- Initial cooling: from \( T_1 = 61^\circ \text{C} \) to \( T_2 = 59^\circ \text{C} \) in \( \Delta t_1 = 10 \) minutes, with room temperature \( T_s = 30^\circ \text{C} \).
- New cooling required: from \( T_3 = 51^\circ \text{C} \) to \( T_4 = 49^\circ \text{C} \).

Using the proportionality from Newton's law:
\[
\frac{\Delta t_1}{\Delta t_2} = \frac{\text{Average Temp Difference in Initial Cooling}}{\text{Average Temp Difference in New Cooling}}
\]

Calculate average temperature differences:
1. For initial cooling: \( \frac{61 + 59}{2} - 30 = 60 - 30 = 30 \)
2. For new cooling: \( \frac{51 + 49}{2} - 30 = 50 - 30 = 20 \)

Then:
\[
\frac{10}{\Delta t_2} = \frac{30}{20}
\]

Solving for \( \Delta t_2 \):
\[
\Delta t_2 = 10 \times \frac{20}{30} = 15 \text{ minutes}
\]