use newton's law of cooling to solve the problem

Question

A dish of lasagna baked at 375 degrees is taken out of the oven into a kitchen that is 72 degrees. After 7 minutes, the temperature of the lasagna is 309.2 degrees. What will its temperature be 17 minutes after it was taken out of the oven? Round your answer to the nearest degree.

A) 239

B) 227

C) 246

D) 234

Answer 1

Hi Adrian,

Thanks for your question!

Newton's law of cooling states

dT/dt = -k*(T-T₀),

where T(t) is the temperature (measured in degF) of the lasagna at time t (measured in minutes), k is a positive constant related to the thermal conductivity of the lasagna (how fast/slow the lasagna cools), and T₀ is the temperature of the kitchen (which is 72 F according to this problem).

This differential equation is separable, meaning we can write the equation as follows:

dT/(T-T₀) = -k*dt.

The LHS just depends on T (since T₀ is constant) and the RHS just depends on t (since k is constant). We can integrate both sides of this equation to get

∫dT/(T-T₀) = ∫-k*dt + C,

ln(T-T₀) = -kt + C,

where C is an arbitrary constant of integration. Solving for T, we get

T(t) = T₀ + e^{-kt + C}.

Since C is an arbitrary constant, we can redefine a new arbitrary constant D = e^C, so that our solution above becomes

T(t) = T₀ + De^-kt.

We know that T₀ = 72 F. It remains to be seen what D and k equal.

Since the lasagna is taken out of an oven with temperature 375 F, we can assume the lasagna has an initial temperature of 375 F. In other words, T(0) = 375 F. Plugging t = 0 into our formula for T(t) above, we find

T(0) = T₀ + De^-k(0)

375 = 72 + D,

so that D = 303 F. Now, our formula for T(t) is

T(t) = 72 + 303e^-kt.

We still don't know the value of k in this formula. To compute this value, we use the fact that the temperature of the lasagna after 7 minutes is 309.2 F. In other words, T(7) = 309.2 F. Plugging t = 7 into our formula for T above, we get

T(7) = 72 + 303e^-k(7),

309.2 = 72 + 303e^-7k.

Solving for k, we find

k = -ln(237.2/303)/7 ∼ 0.0349755912 min^-1.

Our updated formula for T(t) is now

T(t) = 72 + 303e^{-(0.0349755912)t}.

To finish this question, we must determine the temperature of the lasagna at t = 17 min, so we compute

T(17) = 72 + 303 303e^{-(0.0349755912)(17)} ∼ 239 F.

Thus, the final answer is A. The lasagna is roughly 239 F after 17 min.