a body cools from 60 degree celsius to 50 degree celsius in 10 min. if the room temp is 25 degree celsius and assuming newton's laws of cooling to hold good, the temp of the body at the end of next 10 min. will be......

a 38.5 degree celsius

b 40 degree celsius

c 42.85 degree celsius

d 45 degree celsius

To solve this problem, we can apply Newton's Law of Cooling, which 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 difference is small. In this case, we have a body cooling from 60 degrees Celsius to 50 degrees Celsius in 10 minutes, with the room temperature at 25 degrees Celsius. Let's break this down step by step.

Step 1: Understanding the Initial Conditions

The initial temperature of the body, T1, is 60°C, and after 10 minutes, it cools to T2, which is 50°C. The surrounding room temperature, T_room, is 25°C. We can denote the time interval as t = 10 minutes.

Step 2: Applying Newton's Law of Cooling

The formula derived from Newton's Law of Cooling can be expressed as:

T(t) = T_room + (T_initial - T_room) * e^(-kt)

Where:

• T(t) is the temperature of the body at time t.
• T_initial is the initial temperature of the body.
• T_room is the ambient temperature.
• k is the cooling constant.
• e is the base of the natural logarithm.

Step 3: Finding the Cooling Constant (k)

We can first find the value of k using the data we have:

At t = 10 minutes, T(10) = 50°C. Plugging in the values:

50 = 25 + (60 - 25) * e^(-10k)

This simplifies to:

50 - 25 = 35 * e^(-10k)

25 = 35 * e^(-10k)

Now, dividing both sides by 35:

e^(-10k) = 25/35 = 5/7

Taking the natural logarithm of both sides:

-10k = ln(5/7)

Thus, we find:

k = -ln(5/7) / 10

Step 4: Predicting the Temperature After Another 10 Minutes

Now we need to find the temperature after another 10 minutes (t = 20 minutes). We can use the previously calculated k value:

T(20) = 25 + (50 - 25) * e^(-20k)

Substituting the values:

T(20) = 25 + 25 * e^(-20 * (-ln(5/7) / 10))

This simplifies to:

T(20) = 25 + 25 * (5/7)^2

Calculating (5/7)^2 gives approximately 0.5102:

T(20) ≈ 25 + 25 * 0.5102 ≈ 25 + 12.755 = 37.755°C

Final Calculation

So, rounding this to two decimal places, we find that the temperature of the body at the end of the next 10 minutes will be approximately 37.76°C. Among the options provided, the closest answer is:

• a) 38.5 degrees Celsius
• b) 40 degrees Celsius
• c) 42.85 degrees Celsius
• d) 45 degrees Celsius

Thus, the correct answer is option a) 38.5 degrees Celsius.