how long does it take a cup of coffee to cool?
A freshly brewed cup of coffee has temperature 95°C in a 20°C room. When its temperature is 74°C, it is cooling at a rate of 1°C per minute. When does this occur? Approximate to three decimal places.
Thought it might have been newton’s law of cooling but that didn’t work, any help would be much appreciated.
Tagged with: cup of coffee • decimal places • newton
Filed under: Cool Brew Coffee
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yes, use newton’s law of cooling
start with dT/dt = -k(T-Ta) where T is temp, t is time, k is the constant of proportionality and Ta is the ambient temp = 20
separate variables, integrate and get:
T=Ta+Ae^(-kt) where A is the constant of integration
you need to solve for A and k
solve for A knowing that T=95 when t=0:
95=20+A or A=75
solve for k from:
dT/dt = -k(T-Ta)
when T= 74, dT/dt =-1, so k=-1/(74-20) = -1/54=-0.0185
so the total cooling law is T=20+75e^(-0.0185t)
now, when T=75, solve for t:
74=20+75e^(-0.0185t)
ln(54/75)=-0.0185t
t=17.757 mins