Question

The temperature of the body drops from 60°C to 40°C in 7 min. The surrounding temperature is 10°C. The temperature of the body drops from 40°C to T°C in 7 min. Find the value of T

• A. 16
• B. 20
• C. 28
• D. 30

Answer 1

The Correct Option is C

We can solve this problem using Newton's Law of Cooling, which states that the rate of change of temperature of an object is proportional to the difference between its temperature and the ambient temperature.

The formula for Newton's Law of Cooling is:

\(\frac{dT}{dt} = -k(T - T_s)\)

where:

• \(T\) is the temperature of the body at time \(t\)
• \(T_s\) is the surrounding (ambient) temperature
• \(k\) is a constant

Let's consider the two drops:

1. The temperature drops from 60°C to 40°C in 7 minutes. We can write this as:

\(T_1 = 60, T_2 = 40, t_1 = 7 \, \text{min}, T_s = 10\)

From the formula:

\(\frac{40 - 10}{60 - 10} = e^{-kt_1}\)

Simplifying:

\(\frac{30}{50} = e^{-7k}\)

\(\frac{3}{5} = e^{-7k}\) ... (1)

1. Next, the temperature drops from 40°C to \(T\)°C in another 7 minutes.

For this, we write:

\(T_3 = T, t_2 = 7 \, \text{min}\)

\(\frac{T - 10}{40 - 10} = e^{-kt_2}\)

Substituting for the second 7 minutes:

\(\frac{T - 10}{30} = e^{-7k}\)

Using equation (1), we set \(e^{-7k} = \frac{3}{5}\):

\(\frac{T - 10}{30} = \frac{3}{5}\)

Solving for \(T\):

\(T - 10 = 30 \cdot \frac{3}{5}\)

\(T - 10 = 18\)

\(T = 28\)

Thus, the temperature \(T\) for the body after another 7 minutes is 28°C. Therefore, the correct option is 28.