a cup of coffee has cooled from 92 degrees C to 60 degrees C after 12 minutes in a room at 22 degrees C C. How long will it take to cool to 40 degrees?

Question

round to the nearest tenth as needed.

Answer 1

Temperature differences decay exponentially, so if T is the coffee temp and R is the room temp, we have:

T-R = Ae^-bt

plugging in zero for t we get:

92-22=A

A=70

then plug in t=12

60-22=70e^-b12

38=70e^-12b

38/70=e^-12b

take ln() of both sides

ln(38/70)=-12b

b=-ln(38/70)/12

then plug in 40 for T and solve for t

40-22=70e^{ln(38/70)t/12}

18/70=e^{ln(38/70)t/12}

ln(18/70)=ln(38/70)t/12

t=12ln(18/70)/ln(38/70)

t=26.7 min

Answer 2

-32 degrees in 12 min = -32/12 = -16/6 = -8/3 with intercept (0,92) and (12,60)

check:

B = y - mx = 92 - (-8/3)(0) = 92 because (0,92) is the intercept!!!

y = (-8/3)x+92