Determine the temperature of the turkey 45 minutes after it was taken out of the oven given it was 160 degrees Fahrenheit after 30 minutes: Question 7 A roasted turkey is taken from an oven when its temperature has reached $185^{\circ}$ Fahrenheit and is placed on a table in a room where the temperature is $75^{\circ}$ Fahrenheit. Give answers accurate to at least 2 decimal places. (a) If the temperature of the turkey is $160^{\circ}$ Fahrenheit after half an hour, what is its temperature after 45 minutes? Fahrenheit (b) When will the turkey cool to $100^{\circ}$ Fahrenheit? hours.

Step 1 ：Given values are the initial temperature of the turkey \(T0 = 185^{\circ}\) Fahrenheit, the temperature of the surroundings \(Ts = 75^{\circ}\) Fahrenheit, the temperature of the turkey after 30 minutes \(T_{30} = 160^{\circ}\) Fahrenheit, and the time after the turkey was taken out of the oven \(t_{30} = 30\) minutes.

Step 2 ：Calculate the constant of proportionality \(k\) using the formula \(k = -\ln\left(\frac{T_{30} - Ts}{T0 - Ts}\right) / t_{30}\).

Step 3 ：Substitute the given values into the formula to get \(k = -\ln\left(\frac{160 - 75}{185 - 75}\right) / 30 = 0.008594303643403326\).

Step 4 ：Calculate the temperature of the turkey after 45 minutes \(T_{45}\) using the formula \(T_{45} = Ts + (T0 - Ts) \cdot e^{-k \cdot t_{45}}\).

Step 5 ：Substitute the given values and the calculated \(k\) into the formula to get \(T_{45} = 75 + (185 - 75) \cdot e^{-0.008594303643403326 \cdot 45} = 149.71917120428026\).

Step 6 ：The temperature of the turkey 45 minutes after it was taken out of the oven is approximately \(\boxed{149.72}\) degrees Fahrenheit.