Mike records internal temp of a hot sandwich that has been left to cool on a counter. The room temp as An equation is K(t)=60(0.5)t10+2 where K is the temperature in °C and t is the time in mins.

Question

a) What was the temperature of the sandwich when she began to record its temperature?

b) Determine the temperature, to the nearest degree, of the sandwich after 20 min.

c) How much time did it take for the sandwich to reach an internal temperature of 32 °C?

d) What is the domain and range of this function if temperatures cannot be below absolute zero?

Answer 1

The normal cooling equation (Newton) has the form:

Cooling body temperature, K(t)=t_A+(t_H-t_A)e^-kt, or K(t)=(t_H-t_A)e^-kt+t_A, where k is a constant, T_H is the initial temperature of the cooling body (in this case, a sandwich) and T_A is the ambient or room temperature. After a long period the body cools to the ambient temperature. The given equation should resemble this standard equation. t_H-t_A can be replaced by a single quantity which I think is 60 in the given equation. This would mean the initial temperature difference between the internal temperature of the sandwich and the room temperature is 60°C.

e^-kt can be replaced by b^t, that is, a number raised to power x. -kt=ln(b^t)=tln(b), making k=-ln(b). I think that b=0.5 in the given equation. So now we have an explanation for the term 60(0.5)^t. The remaining expression in the given equation is 10+2. This suggests that the ambient temperature T_A is 2°C. That leaves the value 10. I suggest that the given equation is meant to read:

K(t)=60(0.5)^t/10+2, which is an acceptable variation of the standard cooling equation. I think this is the cooling equation. Bearing that in mind we can look at the questions.

(a) When t=0, K(0)=62°C.

(b) K(20)=60(0.5)²+2=17°C.

(c) K(t)=32, 60(0.5)^t/10=30, 0.5^t/10=0.5, so t/10=1, t=10 minutes.

(d) Absolute zero is -273°C approx, but we're not going to need this. The cooling equation indicates that after an indefinitely large time, K=2°C. So the domain is [0,∞) or 0≤t<∞. The range for this domain is K(0)=62°C down to nearly 2°C, that is (2,62] °C or 2<K≤62 (°C).