Kylie and her cousin Aubrey cooked a pizza and when they took it out of the oven it had an initial temperature of 450 degrees fahrenheit

Question

The thermometer at Kylie's house read 75 degrees fahrenheit. After one minute, Kylie took a bite out of the pizza and burnt the roof of her mouth (it was 370 degrees fahrenheit).

a. Given the model T(t)= Ts + (T0-Ts)e^(-kt) and that T(t)= temperature at a given time, t, Ts = temperature of surroundings, T0 = intial temperature, and k=constant, find the exact vaule of k.

b.Write the function that models this situation.

c. What is the temperature of the pizza after 5 minutes?

d. How long must Kylie an Aubrey wait to eat the pizza at a reasonable temperature of 120 degrees fahrenheit?

e. Aubrey decided to place the pizza in the fridge (40 degrees fahrenheit) to cool her pizza. How long will she have to wait to eat it at 120 degrees fahrenheit?

Answer 1

a.

Plugging in the given values:

T(t)=75+(450-75)e^-kt and plugging in t=1 min and T(1)=370=75+375e^-k.

295/375=e^-k, k=-ln(295/375)=0.24 approx.

b.

T(t)=75+(450-75)e^-0.24t and (c) T(5)=75+375e^-1.2=188℉ approx. We may need to replace the numerical values with T₀ and Ts to cover the general case with a more general formula.

d.

When T(t)=120=75+375e^-0.24t, 45/375=e^-0.24t, -0.24t=ln(45/375)=-2.1203, t=8.83 minutes.

e.

T(t)=120=40+(370-40)e^-0.24t, assuming the pizza was refrigerated 1 minute after removing from the oven.

So 80/330=e^-0.24t, t=-ln(80/330)/0.24=5.9 minutes.