One day in the morning,Ramesh filled up $\frac{1}{3}$ of a bucket with hot water from a geyser to take a bath. The remaining $\frac{2}{3}$ was to be filled with cold water (at room temperature) to bring the mixture to a comfortable temperature. Suddenly,Ramesh had to attend to some work that would take $5-10 \text{ min}$ before he could take a bath. He had two options: $(1)$ fill the remaining bucket completely with cold water and then attend to the work,$(2)$ first attend to the work and then fill the remaining bucket just before taking a bath. Which option do you think would have kept the water warmer? Explain.

(B) Option $(2)$ will keep the water warmer.
According to Newton's law of cooling,the rate of cooling is directly proportional to the temperature difference between the substance and its surroundings,i.e.,$\frac{dT}{dt} \propto (T - T_s)$.
In option $(1)$,the bucket is filled with a mixture of hot and cold water. This mixture has a higher average temperature compared to the surrounding room temperature,leading to a higher rate of heat loss to the surroundings during the $5-10 \text{ min}$ delay.
In option $(2)$,the bucket contains only the hot water (from the geyser) for the duration of the delay. While the hot water itself cools down,the total heat capacity of the system is different,and more importantly,by adding the cold water only at the end,we minimize the time the final mixture spends at a higher temperature relative to the surroundings. Therefore,option $(2)$ results in a warmer bath.