Question

A roasted turkey is taken from an oven when its temperature has reached 185°F and is placed on a table in a room where the temperature is 53°F

(a) If the temperature of the turkey is 150°F after half an hour, what is its temperature after 45 min? (Round your answer to the nearest whole number.)

(b) When will the turkey cool to 100°F? (Round your answer to the nearest whole number.)

Answer 1

The temperature will drop on an exponential decay curve, i.e. a curve with the equation
F = 53 + (185-53)exp(-a.t) = 53 + 132.exp(-a.t)
for some parameter a which we need to determine. We know that because at t = 0, F = 185, and as t increases without limit, F tends to 53. We know from (a) that 150 = 53 + 132.exp(-30a), so -30a = ln((150-53)/(185-53)) = -0.05239, so a = 0.01027. So the temperature after 45 mins should be:
53 + 132.exp(-0.01027 x 45) = 136 degrees to the nearest degree.
The turkey cools to 100 degrees F when 100 = 53 + 132.exp(-0.01027 x t)
and that will be when ln((100-53)/132) = -0.01027 x t,
i.e. t = -ln((100-53)/132) / 0.01027 = 101 minutes to the nearest minute.