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between the temperature of the coﬀee and its surroundings. Note in Equation 6.6 that the constant
k is positive for both the cooling and warming scenarios. What determines if the function T (t)
is increasing or decreasing is if T0 (the initial temperature of the object) is greater than Ta (the
ambient temperature) or vice-versa, as we see in our next example.
Example 6.5.4. A 40◦ F roast is cooked in a 350◦ F oven. After 2 hours, the temperature of the
roast is 125◦ F.
1. Assuming the temperature of the roast follows Newton’s Law of Warming, ﬁnd a formula for
the temperature of the roast T as a function of its time in the oven, t, in hours.
2. The roast is done when the internal temperature reaches 165◦ F. When will the roast be done?
1. The initial temperature of the roast is 40◦ F, so T0 = 40. The environment in which we
are placing the roast is the 350◦ F oven, so Ta = 350. Newton’s Law of Warming tells us
T (t) = 350 + (40 − 350)e−kt , or T (t) = 350 − 310e−kt . To determine k...
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