How Long Before The Temperature is Below 90 Degrees?

Question

Coffee is poured at 145ºF into a travel cup. After 25 minutes in a 70º room the coffee has cooled to 137ºF. I accidentally left the coffee on the counter when I left. What is the temperature after 4 hours? after 5 hours? how long before the temperature is below 90º?

Show your process.

Answer 1

I suspect that there is a formula given with this problem, or an equation was talked about as this problem involves Newton's Law of Cooling which is given by

y(t) = T + (y₀-T)e^-kt

where y(t) is the temperature of a body at time t, T is the ambient temperature, y₀ is the initial temperature of the body (ie. temp at t=0), and k is a constant of proportionality. In this problem, let's take t to be measured in minutes and y(t) measured in *F. So the givens are T=70*, and y₀=145* Plugging these into the formula, we have

y(t) = 70 + 75e^-kt

Observe that we are not given the value of k. We must compute it. But how? Well, we are given y(25)=137. So

137 = 70 + 75e^-25k

Solving the equation for k yields k = -(1/25) ln(67/75). So

y(t) = 70 + 75e^{(1/25)·ln(67/75)t}

We want to know the temperature of the coffee after 4 hours. However, we cannot just plug 4 in for t because at the start, we specified that t is measured in minutes! So we convert 4 hours to 240 minutes, and plug this into the formula and compute.

y(240) = 70 + 75e^{(1/25)·ln(67/75)·240}

Likewise to find the temperature after 5 hours.

To find how long it takes before the temp drops below 90*F, we must find the value of t which satisfies y(t)=90, which means solving the equation for t.