A roasted turkey is taken from an oven when its temperature has reached 185°F and is placed on a table in a room where the temperature is 53°F
(a) If the temperature of the turkey is 150°F after half an hour, what is its temperature after 45 min? (Round your answer to the nearest whole number.)
(b) When will the turkey cool to 100°F? (Round your answer to the nearest whole number.)
assuming steady change in temperature, since it changed -35deg/30min, so after m minutes,
T(m) = 185 - 7/6 m
T(45) = 185 - 7/6 * 45 = 132.5
100 = 185 - 7/6 m
m = 73 min
This is a nice exercise, but the real formula for temperature change is not quite so simple.
where did you ge the 7/6?
the temperature dropped 35 deg in 30 min = 35/30 deg/min = 7/6 deg/min
To solve this problem, we need to understand the concept of heat transfer and use the equation for cooling or heating with a constant rate.
The equation for cooling or heating with a constant rate is given by:
𝑇(𝑡) = 𝑇₀ + (𝑇ₑ − 𝑇₀)𝑒^(−𝑘𝑡)
Where:
𝑇(𝑡) represents the temperature at time 𝑡
𝑇₀ represents the initial temperature
𝑇ₑ represents the temperature of the environment
𝑘 is a constant that determines the rate of cooling or heating
Now let's break down each part of the problem:
(a) If the temperature of the turkey is 150°F after half an hour, what is its temperature after 45 minutes?
We have the following values:
Initial temperature (𝑇₀) = 185°F
Temperature of the environment (𝑇ₑ) = 53°F
Time elapsed (𝑡) = 30 minutes
From the equation, we need to find the value of 𝑇(𝑡). So plug in these values into the equation and solve:
𝑇(𝑡) = 𝑇₀ + (𝑇ₑ − 𝑇₀)𝑒^(−𝑘𝑡)
150 = 185 + (53 − 185)𝑒^(−𝑘(30))
To solve for 𝑘, rearrange the equation:
𝑒^(−𝑘(30)) = (150 − 185) / (53 − 185)
Take the natural logarithm of both sides:
−𝑘(30) = ln((150 − 185) / (53 − 185))
Solve for 𝑘:
𝑘 = ln((185 - 150) / (185 - 53)) / 30
Now that you have the value of 𝑘, plug it back into the equation, along with the new time value of 45 minutes, and solve for 𝑇(𝑡):
𝑇(𝑡) = 𝑇₀ + (𝑇ₑ − 𝑇₀)𝑒^(−𝑘𝑡)
(b) When will the turkey cool to 100°F?
We have the following values:
Initial temperature (𝑇₀) = 185°F
Temperature of the environment (𝑇ₑ) = 53°F
Target temperature (𝑇(𝑡)) = 100°F
From the equation, we need to find the value of 𝑡. So plug in these values and solve for 𝑡:
100 = 185 + (53 − 185)𝑒^(−𝑘𝑡)
To solve for 𝑡, rearrange the equation:
𝑒^(−𝑘𝑡) = (100 − 185) / (53 − 185)
Take the natural logarithm of both sides:
−𝑘𝑡 = ln((185 - 100) / (185 - 53))
Solve for 𝑡:
𝑡 = ln((185 - 100) / (185 - 53)) / 𝑘
Now you can substitute the value of 𝑘 from part (a) into this equation to find the time it takes for the turkey to cool to 100°F.