A whole roast chicken is taken from the oven when its temperature reached 185 degrees Fahrenheit. It is then placed on a table where the room temperature is 75 degrees Fahrenheit. After thirty minutes, the temperature of the viand is 150 degrees Fahrenheit. After 45 minutes more, what is the temperature of the roast chicken (in Fahrenheit)?
Ts = T surrounding
(T(t) -Ts)/ (To-Ts) = e^-kt
ln [ (T(t) -Ts)/ (To-Ts)] = - k T
I would convert those F temps to C and add 273 to get absolute temps
Since the temps were given in Fahrenheit, there's no real reason to convert to Celsius or Kelvin. Whichever scale you use, it just means a different value for k.
To find the temperature of the roast chicken after 45 more minutes, we can use Newton's Law of Cooling, which states that the rate of change of temperature of an object is directly proportional to the difference between its temperature and the surrounding temperature.
First, let's calculate the initial temperature of the roast chicken.
The temperature of the viand after 30 minutes is 150 degrees Fahrenheit.
Using Newton's Law of Cooling, the rate of change of temperature is proportional to the difference between the temperature of the viand and the room temperature.
The temperature difference is (150 - 75) = 75 degrees Fahrenheit.
Now, let's calculate the constant proportionality.
We know that the temperature of the roast chicken at the oven was 185 degrees Fahrenheit, and after 30 minutes it cooled down to 150 degrees Fahrenheit.
Using the exponential decay formula:
T = T0 + (Ts - T0) * e^(-kt)
where T is the temperature at time t, T0 is the initial temperature, Ts is the surrounding temperature, and k is the constant proportionality.
Substituting the values into the equation:
150 = 185 + (75 - 185) * e^(-30k)
-35 = -110 * e^(-30k)
e^(-30k) = -35 / -110
e^(-30k) = 7/22
Taking the natural logarithm of both sides to solve for k:
-30k = ln(7/22)
k = ln(7/22) / -30
k ≈ -0.0481
Now, let's use the equation to find the temperature after 45 more minutes.
T = Ts + (T0 - Ts) * e^(-kt)
where T is the temperature at time t, T0 is the initial temperature, Ts is the surrounding temperature, and k is the constant proportionality.
T = 75 + (150 - 75) * e^(-0.0481 * 45)
T = 75 + 75 * e^(-0.0481 * 45)
T ≈ 75 + 75 * e^(-2.1655)
T ≈ 75 + 75 * 0.1148
T ≈ 75 + 8.61
T ≈ 83.61
Therefore, the temperature of the roast chicken after 45 more minutes is approximately 83.61 degrees Fahrenheit.
To find the temperature of the roast chicken after 45 minutes, we need to understand how it loses heat over time.
We know that the temperature of the roast chicken at the start is 185 degrees Fahrenheit when it is taken out of the oven. After 30 minutes, the temperature drops to 150 degrees Fahrenheit.
We can assume that the roast chicken loses heat at a constant rate. This means that for every unit of time, the temperature decreases by the same amount.
To find the rate at which the roast chicken loses heat, we subtract the final temperature from the initial temperature and divide it by the time it took for the temperature to drop.
Rate of temperature decrease = (Initial temperature - Final temperature) / Time taken
Rate of temperature decrease = (185 - 150) / 30
Rate of temperature decrease = 35 / 30
Rate of temperature decrease = 1.17 degrees Fahrenheit per minute (approximately)
Now, to find the temperature of the roast chicken after 45 minutes, we need to multiply the rate of temperature decrease by the time elapsed and subtract it from the initial temperature.
Temperature after 45 minutes = Initial temperature - (Rate of temperature decrease x Time)
Temperature after 45 minutes = 185 - (1.17 x 45)
Temperature after 45 minutes ≈ 185 - 52.65
Temperature after 45 minutes ≈ 132.35 degrees Fahrenheit
Therefore, the temperature of the roast chicken after 45 minutes is approximately 132.35 degrees Fahrenheit.