Newton's Law of Cooling says that the rate at which a body cools is proportional to the difference in temperature between the body and an environment into which it is introduced. Suppose hot water at a temperature of 175 degrees Fahrenheit is poured into a mug where the room temperature is 70 degrees Fahrenheit. Then, 9 minutes later the temperature of the water is 130 degrees Fahrenheit. Use Newton's law of cooling to find temperature, T, of the water t minutes after it is poured into the mug. Round any constants to four decimal places.

Question

Newton's Law of Cooling says that the rate at which a body cools is proportional to the difference in temperature between the body and an environment into which it is introduced. Suppose hot water at a temperature of 175 degrees Fahrenheit is poured into a mug where the room temperature is 70 degrees Fahrenheit. Then, 9 minutes later the temperature of the water is 130 degrees Fahrenheit. Use Newton's law of cooling to find temperature, T, of the water t minutes after it is poured into the mug. Round any constants to four decimal places.
(a)  T(t)=
(b) What is the temperature in 35 minutes? Temperature = 
 degrees Fahrenheit
(c) What is the rate at which the temperature in changing in 35 minutes?  
∘F/min

oobleck · Accepted Answer

(a) Recall that T(t) = Te + (Te-To)e^(-kt)
T(t) = 70 + (175-70)e^(-kt) = 70 + 105e^(-kt)
since T(9) = 130, find k by solving
70 + 105e^(-9k) = 130
k = 1/9 log(7/4) = 0.062
and so
T(t) = 70 + 105e^(-0.062t)

(b) find T(35)
(c) dT/dt = -6.51 e^(-0.062t)
so find T'(35)

susu · Answer

Thanks, 100% correct!