Answer :
Final answer:
Using Newton's Law of Cooling, we can determine the temperature of a roast turkey as it cools down in a room. If the temperature of the turkey is 150 degrees Fahrenheit after half an hour, it will be approximately 111.15 degrees Fahrenheit after 45 minutes. The turkey will cool to 100 degrees Fahrenheit after approximately 44 minutes and 53 seconds.
Explanation:
(a) First, we can determine the rate at which the turkey cools down by using Newton's Law of Cooling: ΔT = k(T_m - T_s), where ΔT is the change in temperature, k is the cooling constant, T_m is the temperature of the surrounding medium, and T_s is the temperature of the object. In this case, ΔT = 150 - 75 = 75 degrees Fahrenheit, T_m = 75 degrees Fahrenheit, and T_s = 185 degrees Fahrenheit. Plugging these values into the formula, we get 75 = k(75 - 185), which simplifies to k = -0.5.
To find the temperature after 45 minutes, we can use the formula: T = T_s + ΔT * e^(kt), where T_s is the starting temperature, ΔT is the change in temperature, k is the cooling constant, and t is the time in minutes. Plugging in the values, we have T = 185 + 75 * e^(-0.5 * 45). Calculating this, we find that the temperature after 45 minutes is approximately 111.15 degrees Fahrenheit.
(b) To find when the turkey will cool to 100 degrees Fahrenheit, we can use the formula from part (a). Plugging in the values, we have 100 = 185 + 75 * e^(-0.5 * t). Solving for t, we find t = -ln((100 - 185) / 75) / -0.5. Calculating this, we find that the turkey will cool to 100 degrees Fahrenheit after approximately 44 minutes and 53 seconds.
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