Question

An object with initial temperature 130 ∘ F is submerged in large tank of water whose temperature is 50 ∘ F . Find a formula for F ( t ) , the temperature of the object after t minutes, if the cooling constant is k = − 0.2 . Remember Newton's Law of Cooling (the rate of change of temperature with respect to time is equal to k times the difference between the temperature of the object and the surrounding temperature) ! :)

Answer 1

Answer:

T = 50 + 80e^(-0.2t)

Explanation:

Newton's law of cooling says the rate of change of temperature with respect to time is proportional to the temperature difference:

dT/dt = k (T − Tₐ)

Separating the variables and integrating:

dT / (T − Tₐ) = k dt

ln (T − Tₐ) = kt + C

T − Tₐ = Ce^(kt)

T = Tₐ + Ce^(kt)

Given that Tₐ = 50 and k = -0.2:

T = 50 + Ce^(-0.2t)

At t = 0, T = 130.

130 = 50 + Ce^(0)

130 = 50 + C

C = 80

Therefore:

T = 50 + 80e^(-0.2t)