To solve this problem, we need to analyze the cooling of body X according to Newton's Law of Cooling and then consider the heat transfer between body X and the large box Y through the conducting rod. Let's break this down step by step.

Understanding Newton's Law of Cooling

Newton's Law of Cooling states that the rate of heat loss of a body is directly proportional to the difference in temperature between the body and its surroundings, provided this temperature difference is small. Mathematically, it can be expressed as:

Q = hA(T - T_a)

Where:

• Q = heat lost
• h = heat transfer coefficient
• A = surface area of the body
• T = temperature of the body
• T_a = ambient temperature

Cooling of Body X

Initially, at time t=0, the temperature of body X is T_0. As it cools, at time t, its temperature is T_x. According to Newton's Law of Cooling, we can express the temperature of body X over time as:

T_x = T_a + (T_0 - T_a)e^{-kt}

Here, k is a constant that depends on the properties of the body and the surrounding medium. Given that at a certain time, the temperature of body X is found to be 350 K, we can denote this as:

T_x = 350 K

Heat Transfer to Box Y

Now, when body X is connected to box Y through a conducting rod, heat will flow from body X to box Y until thermal equilibrium is reached. The heat transfer through the rod can be described by Fourier's Law of Heat Conduction:

Q = \frac{KA(T_x - T_y)}{L}

Where:

• K = thermal conductivity of the rod
• T_y = temperature of box Y (assumed to be T_a)
• L = length of the rod

Finding the Temperature of Body X

At the moment when the temperature of body X is 350 K, we can set up the equation for heat transfer:

Q = C \frac{dT_x}{dt} = \frac{KA(T_x - T_a)}{L}

Since the heat capacity of box Y is very large, we can assume that its temperature does not change significantly. Thus, we can set T_y = T_a. Rearranging the equation gives us:

C \frac{dT_x}{dt} = \frac{KA(T_x - T_a)}{L}

To find the temperature of body X at a later time, we can integrate this equation over time, but for simplicity, we can analyze the steady state where the heat loss from body X equals the heat gain by box Y. This leads us to:

T_x = T_a + \frac{KA}{CL}(T_x - T_a)

Solving this equation will yield the temperature of body X at the time of interest. However, since we need a specific numerical answer, we would need the values of C, K, A, L, and T_a to compute T_x accurately.

Conclusion

In summary, the temperature of body X can be determined by applying Newton's Law of Cooling and Fourier's Law of Heat Conduction. By analyzing the heat transfer between body X and box Y, we can derive the temperature of body X at any given time, provided we have the necessary parameters. If you have specific values for these parameters, we can plug them in to find the exact temperature of body X.