Part 4/7:
Suppose you initially stuff a turkey at a temperature of 50 degrees Fahrenheit and place it in a 325-degree oven. After one hour, the temperature of the turkey is recorded to be 93 degrees Fahrenheit, and after two hours, its temperature is 129 degrees. To predict the turkey's temperature after three hours, we can use linear approximation.
First, we plot the temperature against time and mark the known temperatures at 0, 1, and 2 hours. To find the linear approximation around the point ( t = 2 ), we can derive the slope (( f'(2) )) using:
[
f'(2) \approx \frac{f(2) - f(1)}{2 - 1} = \frac{129 - 93}{1} = 36
]
Now using the linear approximation formula, we have:
[
L(t) = 36(t - 2) + 129
]
To predict the temperature at ( t = 3 ):
[