This exercise uses Newtons Law of Cooling A roasted turkey i

This exercise uses Newton\'s Law of Cooling. A roasted turkey is taken from an oven when its temperature has reached 185 degree F and is placed on a table in a room where the temperature is 73 degree F. (a) If the temperature of the turkey is 155 degree F after half an hour, what is its temperature after 45 min? (Round your answer to the nearest whole number.) degree F (b) After how many hours will the turkey cool to 100 degree F? hr

Solution

formula for newton\'s law of cooling is T(t)=T_s +e^-kt(T_o-T_s)

where  T(t) is temperator of the turkey after time t , k is constant ,T_s =room temperature , T_o =initial temperature of the turkey

(a)

T(t)=73+e^-kt(185-73)

T(t)=73+ 112e^-kt

after half an hour , t =30_/60=0.5 hour

=>73+ 112e^-k*0.5 =155

=>e^-k*0.5(112) =82

=>e^0.5k=112_/82

=>0.5k=ln(112_/82)

=>k=2*ln(112_/82)

=>k=0.62356

T(t)=73+ 112e^-0.62356t

after 45 min, t=45_/60 =0.75 hour

T=73+ 112e^{-0.62356*0.75}

T=73+70.1636

T=143

temperature of turkey is 143^oF after 45 min

(b)

T(t)=100

=>73+ 112e^-0.62356t=100

=>112e^-0.62356t=27

=>e^0.62356t=112/27

=>t=(1_/0.62356)ln(112_/27)

=>t=2.28

it takes 2.3 hours for turkey to cool to 100^oF