1 pt) Consider a 42 F object placed in 70 Froom. (a) Write a differential equati
ID: 2869735 • Letter: 1
Question
1 pt) Consider a 42 F object placed in 70 Froom. (a) Write a differential equation for H, the temperature of the object at timet, using k 0 for the constant of proportionality, and write your equation in terms of H,k, and t. (b) Give the general solution for your differential equation. Simplify your solution in terms of an unspecified constant C, which appears as the coefficient of an exponential term, and the growth factor k. (Your answer may involve the constant of proportionality k.) (c) The temperature of the object is 42 F initially, and 50 F one hour later. Find the temperature of the object after 3 hours. H(3) degrees FExplanation / Answer
(1)
Newton's Law of cooling: the rate of temperature change of an object, H(t), is proportional to the difference between the temperature of the surroundings and the temperature of the object:
dH(t)/dt = -k*(H(t) - H_surr)
where k is a positive constant of proportionality.
This is a separable differential equation:
dH/(H(t) - H_surr) = -k dt
(2)
Integrating (assuming that the temperature of the surroundings is constant):
ln(H(t) - H_surr) - ln(Ho - H_surr) = -k*t
where Ho is the temperature of the object at time t = 0.
ln((H(t) - H_surr)/(Ho - H_surr)) = -k*t
H(t) = H_surr + (Ho - H_surr)*exp(-k*t) = H_surr + C*exp(-k*t)
This is the general solution to the differential equation. The constant "C" that you are supposed to use in your solution, is, in fact, the difference between the initial temperature of the object and the constant temperature of the surroundings.
(3)
In this case:
H(t) = 70
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