Answer:
Newton’s Law of Cooling states that the rate of change of the temperature of an object is proportional to the temperature difference between the object and its surroundings. This can be modeled by the differential equation dT/dt = k(T - Ta), where T is the temperature of the object after t units of time have passed, Ta is the ambient temperature of the object’s surroundings, and k is a constant of proportionality.
Suppose that a cup of coffee begins at 95 degrees and, after sitting in room temperature of 25 degrees for 15 minutes, the coffee reaches 70 degrees. We can use this information to solve for the constant of proportionality k. The solution to the differential equation is given by T(t) = Ta + (T(0) - Ta)e^(kt), where T(0) is the initial temperature of the coffee. Plugging in the values we have, we get:
70 = 25 + (95 - 25)e^(15k) 45 = 70e^(15k) e^(15k) = 45/70 15k = ln(45/70) k = ln(45/70)/15
Now that we have solved for k, we can use the solution to find how long it will take before the coffee reaches 50 degrees. Plugging in the values we have, we get:
50 = 25 + (95 - 25)e^(kt) 25 = 70e^(kt) e^(kt) = 25/70 kt = ln(25/70) t = ln(25/70)/k
Plugging in the value we found for k, we get:
t = ln(25/70)/(ln(45/70)/15) ≈ 30.2
So it will take approximately 30.2 minutes for the coffee to reach 50 degrees.
Received message. Newton's Law of Cooling states that the rate of change of the temperature of an object is proportional to the temperature difference between the object and its surroundings. This can be modeled by the differential equation dT/dt = k(T - Ta), where T is the temperature of the object after t units of time have passed, Ta is the ambient temperature of the object's surroundings, and k is a constant of proportionality. Suppose that a cup of coffee begins at 95 degrees and, after sitting in room temperature of 25 degrees for 15 minutes, the coffee reaches 70 degrees. We can use this information to solve for the constant of proportionality k. The solution to the differential equation is given by T(t) = Ta + (T(0) - Ta)e^(kt), where T(0) is the initial temperature of the coffee. Plugging in the values we have, we get: 70 = 25 + (95 - 25)e^(15k) 45 = 70e^(15k) e^(15k) = 45/70 15k = ln(45/70) k = ln(45/70)/15 Now that we have solved for k, we can use the solution to find how long it will take before the coffee reaches 50 degrees. Plugging in the values we have, we get: 50 = 25 + (95 - 25)e^(kt) 25 = 70e^(kt) e^(kt) = 25/70 kt = ln(25/70) t = ln(25/70)/k Plugging in the value we found for k, we get: t = ln(25/70)/(ln(45/70)/15) ≈ 30.2 So it will take approximately 30.2 minutes for the coffee to reach 50 degrees.