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Newton's formula of cooling or heating
The formula is:
T(t) = S + (T(0) - S)*e^kt
where:
t = time in hours
T(0) = initial temperature of the turkey
T(t) = final temperature of the turkey after t hours.
S = temperature of the surrounding medium.
k = constant rate of cooling or heating.
In your problem:
t = 4 hours
T(0) = 40 degrees for the turkey
T(t) = 170 degrees of the turkey after 4 hours in the oven.
S = temperature of surrounding medium = 350 degree oven.
k = constant of cooling or heating which is what you want to find.
We plug these values into the formula as shown below and solve for k.
T(t) = S + (T(0)-S)*e^kt becomes:
170 = 350 + (40-350) * e^4k
This becomes:
170 = 350 - (310 * e^4k)
subtract 350 from both sides of the equation to getr:
170 - 350 = -310 * e^(4k)
Simplify to get:
-180 = -310*e^(4k)
Divide both sides of the equation by -310 to get:
-180/-310 = e^(4k)
Take natural log of both sides to get:
ln(-180/-310) = ln(e^(4k))
This becomes:
ln(-180/-310) = 4k * ln(e)
Since ln(e) = 1, this becomes:
ln(-180/-310) = 4k
Divide both sides of this equation by 4 to get:
ln(-180/-310) / 4 = k
Solve for k to get:
k = -.135903862
Plug this back into the equation to get:
170 = 350 + (40-350) * e^4*-.135903862
Solve this to get:
170 = 170
Now that we have a value for k, we can start all over with the formula to find out the remaining time.
The formula is:
T(t) = S + (T(0)-S)*e^kt
where:
t = time in hours
T(0) = initial temperature of the turkey
T(t) = final temperature of the turkey after t hours.
S = temperature of the surrounding medium.
k = constant rate of cooling or heating.
We have:
t = time in hours we want to find.
T(0) = initial temperature of the turkey = 170 degrees.
T(t) = final temperature of the turkey after t hours = 185 degrees.
S = temperature of the surrounding medium = 350 degrees of the oven.
k = constant rate of cooling or heating = -.135903862
Formula becomes:
185 = 350 + (170-350)*e^-.135903862*t
We solve for t to get:
-165 = -180*e^-.135903862*t
Divide both sides of this equation by -180 to get:
-165/-180 = e^-.135903862*t
Take natural log of both sides of this equation to get:
ln(-165/-180) = ln(e^-.135903862*t)
Because log(b^n) = n*log(b), this becomes:
ln(-165/-180) = -.135903862*t*ln(e)
Because ln(e) = 1, this becomes:
ln(-165/-180) = -.135903862*t
Divide both sides by -.135903862 to get:
t = ln(-165/-180)/-.135903862
solve for t to get:
t = .640242124 hours
That should be your answer.
To confirm this calculation is good, I went back to the beginning and plugged 4.640242124 hours into the original equation to get:
T(t) = S + (T(0) - S)*e^kt becomes:
185 = 350 + (40-350)*e^4.640242124*-.135903862
Solving the right side of the equation, I get:
185 = 350 + (40-350)*e^4.640242124*-.135903862 = 185.
This confirms the answer is good because we went all the way from 40 degrees to 185 degrees without stopping at 170 degrees and the total time becomes the same as if we went from 40 to 170 and then 170 to 185.
Your answer is:
t = .640242124 hours to go from 170 degrees to 185 degrees.
