Introduction:
In this problem, we are given that a body cools from temperature 3T to 2T in 10 minutes, where T represents the room temperature. We are asked to determine the temperature of the body at the end of the next 10 minutes, assuming Newton's law of cooling is applicable.

Newton's Law of Cooling:
Newton's law of cooling states that the rate of change of temperature of an object is proportional to the difference between its temperature and the ambient temperature. Mathematically, it can be expressed as:

dT/dt = -k(T - T0)

Where dT/dt is the rate of change of temperature, T is the temperature of the object, T0 is the ambient temperature, and k is the cooling constant.

Calculating the Cooling Constant:
To determine the cooling constant, we can use the given information that the body cools from temperature 3T to 2T in 10 minutes. Let's denote the initial temperature as T1 and the final temperature as T2. We can set up the following equation:

dT/dt = -k(T - T0)

(2T - T0) = -(3T - T0)e^(-10k)

Simplifying this equation, we get:

e^(-10k) = (2T - T0)/(3T - T0)

Taking the natural logarithm on both sides, we have:

-10k = ln((2T - T0)/(3T - T0))

Solving for k, we get:

k = -ln((2T - T0)/(3T - T0))/10

Calculating the Temperature at the End of the Next 10 Minutes:
Now that we have determined the cooling constant, we can use it to calculate the temperature of the body at the end of the next 10 minutes. Let's denote the final temperature at the end of the first 10 minutes as T_f1 and the final temperature at the end of the next 10 minutes as T_f2.

Using Newton's law of cooling, we have:

dT/dt = -k(T - T0)

Integrating both sides of the equation, we get:

∫dT/(T - T0) = -k∫dt

ln|T - T0| = -kt + C

Solving for T, we have:

T - T0 = e^(-kt + C)

T = T0 + e^(-kt + C)

Using the initial condition that the body cools from temperature 3T to 2T in 10 minutes, we can determine the value of C. Plugging in the values, we have:

2T = T0 + e^(-k*10 + C)

3T = T0 + e^(-k*0 + C)

Solving these equations simultaneously, we can find the values of C and T_f1. Then, we can substitute the value of T_f1 into the equation T = T0 + e^(-kt + C) to calculate T_f2, the temperature at the end of the next 10 minutes.

Conclusion:
By applying Newton's law of cooling and solving the given problem step by step, we can determine the temperature of the body at the end of the next 10 minutes.