The temperature of a cup of coffee varies according to Newton's Law of Cooling

Question

The temperature of a cup of coffee varies according to Newton's Law of Cooling: , where T is the temperature of the coffee, A is the room temperature, and k is a positive constant. If the coffee cools from 100°C to 80°C in 1 minute at a room temperature of 22°C, find the temperature, to the nearest degree Celsius of the coffee after 4 minutes. 

A)20

B)24

C)46

D)54

Answer 1

dT/dt = -k(T-A)

solve the differential equation

T = A + (T-A)e^-kt

80 degrees after 1 second

plug in the values for T and t

80 = 22 + 78e^-k(1)

58= 78e^-k

58/78 = e^-k

ln(58/78) =-k = ln(29/39)

-k= slightly less than 0.3

T(4) = 22 + 78e^4ln(29/39)

= about 22+23.8 = 45.8 degrees after 4 seconds

Given the choices, 46 is closest.