Exponential Growth and Decay? yakov59 asked 6 years ago A) A cup of coffee has a temperature 95degree Celsisus and takes 30 minutes to cool to 61 degrees celsisus in a room with temperature 20 degrees celsisus. Show that the temperature of the coffee after t minutes is

T(t) = 20 + 75e^-kt K=0.02

B) What is the average temperature of the coffee during the first half hour?

A) According to Newton's Law of Cooling dT/dt = -k(T - Tr) where Tr = 20 (room temperature) d(T - Tr)/(T - Tr) = -kdt ln(T - Tr) = -kt + lnC T - Tr = Ce^(-kt) when t = 0, T = 95, so 95 - 20 = C = 75 T = Tr + Ce^(-kt) T = 20 + 75e^(-kt) to determine k, we use that when t = 30, T = 61 61 = 20 + 75e^(-30k) 41/75 = e^(-30k) k = ln(75/41)/30 = 0.02 therefore T = 20 + 75e^(-0.02t)

B) the average temperature of the coffee during the first half hour Ta = T(t) dt/30 (int from 0 to 30)

Ta = [20 + 75e^(-0.02t)] dt/30 = [20t - 75/0.02 * e^(-0.02t)] /30 = [20*30 - (75/0.02)*(e^(-0.02*30) - 1)]/30 = 76.4 degrees Celsius.

A freshly brewed cup of coffee has temperature 95C in a 20C room. When its temperature is 66C, it is cooling at a rate of 1C per minute. When does this occur? Approximate to three decimal places.

i would be able to do this if the gave me a temperature of the coffee at a particluar time, so i could find k. but i dont know how to do it because i can't find k or t. help! First, remember the differential equation that defines Newton cooling:

dT/dt = -k(T-Tambient)

Tambient is 20C, and you're given that dT/dt = -1C/min when T=66C. That is:

-1C /min = -k(66C - 20C)

That's enough to solve for k, right there.

k = (1/46) min^-1

Now use the solution to the differential equation above:

T = Tambient + (T(0) - Tambient)e^(-kt)

Setting t=0 when T=95C makes that equation

T = 20C + (75C)e^(-t/46)

Then solve for t when T = 66C. Leaving units off for now: 66 = 20 + 75e^(-t/46) e^(-t/46) = 46/75 t = -46*ln(46/75) = 46*ln(75/46)

That's in minutes because the number 46 is 1/k and we're measuring k in inverse minutes. Use a calculator (or 5-place log tables, if you have them, ) to get those three decimal places.