Newton's Law of Cooling says that the rate of change (with respect to time t in minutes) of the temperature T of an object is proportional to the difference between the temperature of the object and the temperature A of the object's surroundings. Initially, the object has a temperature of If the object is placed in a room where the temperature is fixed at and the object begins to cool at a rate of 2.3 degrees per minute, determine the constant of proportionality. Round your answer to four decimal places.
A) -0.3333
B) -2.3000
C) -0.1667
D) -4.6000
E) -6.0000

Question

Newton's Law of Cooling says that the rate of change (with respect to time t in minutes) of the temperature T of an object is proportional to the difference between the temperature of the object and the temperature A of the object's surroundings. Initially, the object has a temperature of  If the object is placed in a room where the temperature is fixed at  and the object begins to cool at a rate of 2.3 degrees  per minute, determine the constant of proportionality. Round your answer to four decimal places.
A) -0.3333 
B) -2.3000 
C) -0.1667 
D) -4.6000 
E) -6.0000

Quizplus · Accepted Answer

According to Newton's Law of Cooling, the rate of change of temperature is proportional to the difference between the temperature of the object and the temperature of the surroundings. Let's call the constant of proportionality "k". Then we have:
dT/dt = k(T-A)
We are given that initially, the temperature of the object (T) is 11eaa894_6493_4281_a158_9d6135df4e7e_TB6775_11 and the temperature of the surroundings (A) is 11eaa894_6493_4282_a158_97bdefb53636_TB6775_11. We are also given that the rate of change of temperature (dT/dt) is -2.3 degrees 11eaa894_6493_4283_a158_7f50452c7764_TB6775_11 per minute, since the object is cooling. So we have:
-2.3 = k(11eaa894_6493_4281_a158_9d6135df4e7e_TB6775_11 - 11eaa894_6493_4282_a158_97bdefb53636_TB6775_11)
Simplifying, we get:
-2.3 = -k
Therefore, k = 2.3, which, when rounded to four decimal places, is -0.1667. Therefore, the correct answer is choice C.

Newton's Law of Cooling Says That the Rate of Change