Newton\'s law of cooling states that the temperature of an object changes at a r

Question

Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of 195 degrees Fahrenheit when freshly poured, and 2 minutes later has cooled to 182 degrees in a room at 80 degrees, determine when the coffee reaches a temperature of 152 degrees. The coffee will reach a temperature of 152 degrees in minutes.

Explanation / Answer

dT/dt = -k(T-S) where T is current temperature and S =ambient temperature

dT/(T-S) = -k.dt

Solving the differential equation gives

ln(T-S) = -kt + C

T-S = e^(-kt+C)

T(t) = S + e^(-kt+C)

T(t) = S +(To-S)*e^(-kt) where To = initial temperature at t = 0

182 = 80 + (195-80) e^(-2k)

k = 0.0455 min^-1

now

152 = 80 + (195-80)e^(-kt)

e^-kt = 152-80 / 195-80

-kt = ln(72/115)

t = 0.4683 / 0.0455 = 10.29 = 10.3 min

Related Questions

Navigate

• Browse (All)
• Browse N
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.