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A body cools from 60oC to 50oC in 10 minutes. If the room temperature is 25oC and assuming Newton's law of cooling to hold good, the temperature of the body at the end of the next 10 minutes will be
- a)38.5oC
- b)40oC
- c)42.85oC
- d)45oC
Correct answer is option 'C'. Can you explain this answer?
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A body cools from 60oC to 50oC in 10 minutes. If the room temperature ...
Newton's Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. Mathematically, it can be expressed as:
dT/dt = -k(T - Ts)
where dT/dt is the rate of change of temperature with respect to time, T is the temperature of the object, Ts is the temperature of the surroundings, and k is a constant of proportionality.
To solve the given problem, we can use the following steps:
Step 1: Find the value of k
Given that the body cools from 60oC to 50oC in 10 minutes, we can use the formula:
dT/dt = -k(T - Ts)
-10/ln(6/5) = k
where ln denotes the natural logarithm. Solving for k, we get:
k = 0.0231 min^-1
Step 2: Find the temperature of the body after 20 minutes
Using the same formula, we can write:
dT/dt = -k(T - Ts)
dT/dt = -0.0231(T - 25)
Separating variables and integrating both sides, we get:
ln(T - 25) = -0.0231t + C
where C is the constant of integration. To find C, we use the initial condition that the temperature of the body is 50oC at t = 10 minutes:
ln(50 - 25) = -0.0231(10) + C
C = ln(25) - 0.231
Substituting this value of C, we get:
ln(T - 25) = -0.0231t + ln(25) - 0.231
ln(T - 25) = -0.0231t + 3.218
T - 25 = e^(-0.0231t + 3.218)
T = 25 + e^(-0.0231t + 3.218)
At t = 20 minutes, we have:
T = 25 + e^(-0.0231(20) + 3.218)
T = 42.85oC
Therefore, the temperature of the body at the end of the next 10 minutes is 42.85oC, which is option (c).
dT/dt = -k(T - Ts)
where dT/dt is the rate of change of temperature with respect to time, T is the temperature of the object, Ts is the temperature of the surroundings, and k is a constant of proportionality.
To solve the given problem, we can use the following steps:
Step 1: Find the value of k
Given that the body cools from 60oC to 50oC in 10 minutes, we can use the formula:
dT/dt = -k(T - Ts)
-10/ln(6/5) = k
where ln denotes the natural logarithm. Solving for k, we get:
k = 0.0231 min^-1
Step 2: Find the temperature of the body after 20 minutes
Using the same formula, we can write:
dT/dt = -k(T - Ts)
dT/dt = -0.0231(T - 25)
Separating variables and integrating both sides, we get:
ln(T - 25) = -0.0231t + C
where C is the constant of integration. To find C, we use the initial condition that the temperature of the body is 50oC at t = 10 minutes:
ln(50 - 25) = -0.0231(10) + C
C = ln(25) - 0.231
Substituting this value of C, we get:
ln(T - 25) = -0.0231t + ln(25) - 0.231
ln(T - 25) = -0.0231t + 3.218
T - 25 = e^(-0.0231t + 3.218)
T = 25 + e^(-0.0231t + 3.218)
At t = 20 minutes, we have:
T = 25 + e^(-0.0231(20) + 3.218)
T = 42.85oC
Therefore, the temperature of the body at the end of the next 10 minutes is 42.85oC, which is option (c).
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A body cools from 60oC to 50oC in 10 minutes. If the room temperature is 25oC and assuming Newtons law of cooling to hold good, the temperature of the body at the end of the next 10 minutes will bea)38.5oCb)40oCc)42.85oCd)45oCCorrect answer is option 'C'. Can you explain this answer?
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