Consider a 35∘F object placed in 64∘F room.?
Consider a 35∘F object placed in 64∘F room.
Write a differential equation for H, the temperature of the object at time t, using k>0 for the constant of proportionality, and write your equation in terms of H, k, and t.
H' = ?
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.
H = ?
The temperature of the object is 35∘F initially, and 43∘F one hour later. Find the temperature of the object after 3 hours.
H(3) = ? degrees F
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Newton's Law of cooling that 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 - H_surr) = -k dt
Integrating (assuming that the temperature of the surroundings is constant):
ln(H - H_surr) - ln(Ho - H_surr) = -k*t
where Ho is the temperature of the object at time t = 0.
ln((H - 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.
In this case:
H(t) = 64°F + (35°F - 64°F)*exp(-k*t) = 64°F - (29°F)*exp(-k*t)
We are told that after 1 hr, the temperature of the object is 43°F. We can use this information to solve for the constant k:
H(1 hr) = 43°F = 64°F - (29°F)*exp(-k*hr)
21°F = (29°F)*exp(-k*hr)
(1/hr)*ln(21/29) = -k
k = 0.323/hr
So the solution for this case is:
H(t) = 64°F - (29°F)*exp(-0.323*t/hr)
After 3 hours the temperature of the object is:
H(3 hr) = 64°F - (29°F)*exp(-0.323*3) = 53°F