Calculus homework help

Question

1. A roast turkey is taken from an oven when its temperature has reached 185 degrees and is placed on a table in a room where the temperature is 75 degrees. (Round your answer to the nearest whole number). If the temperature of the turkey is 150 degrees after half an hour, what is the temperature after 60 minutes?

2. Find the absolute minimum and maximum values of f on the interval [-1, 3]
f(t) = t • √9-t^2

Answer 1

The general form of an equation for exponential cooling is

T(t) = T_a+(T₀-T_a)e^-kt .

T_a is the final temperature;

T₀ is the initial temperature;

k is the decay constant.

 

We have T₀=185^o, T_a=75^o .

 

We use the condition when t =30 min to find the constant k. T=150^o when the time is 30 min.

 

T(30) = 75 + (185 - 75)e^-k*30

We use the values from the problem to find k.

150 = 75 + 110e^-k*30

75 = 110e^-k*30

 

We take natural logarithm to finish the problem.

ln(75) = ln((110 e^-k30))

 

 

ln(75) = ln(110) +ln(e^-k30)

 

We use the rule that logarithm of e to the power of x is equal to x. In our case we have (-k30) instead.

ln(75) - ln(110) = -30k

k =- [ln(75) - ln(110)]/30

k =0.0127664

 

We use the found value of k in the formula and calculate the temperature after 60 min.

T(60) = 75 + 110 e^-60*.0127664

         =126^o

 

Note, if we measure time in hours, the constant k will change, but the value of the exponent will remain the same. The answer will not change.