This phenomenon is modeled with a differential equation, and that equation may be solved to give
T(t) = TA + [T0 − TA]b^t
where T(t) is the varying temperature of the object at a
given time, t, TA is the surrounding ambient temperature,
T0 is the initial temperature of the object, and b is a
constant that depends on the material the object is
composed of and how fast it heats or cools.
Suppose you decided to make a cup of hot tea
heated to 220 F in the kitchen that is at 70 F.
1.Solve the above equation for the constant b.
2.If the cup of hot tea cooled to 170 F in
10 minutes, find the value of the constant b in
the above equation. Express your answer to
5 decimal places.
3.Solve the above equation for t.
4.Suppose you like your hot tea at the tepid
temperature of 100 F. How long, to the nearest
minute, will you have to wait until it cools to this
temperature?
To go along with your hot tea, you take a frozen apple pie from
the freezer and place it in the oven preheated to 350 F. Assume the
freezer is at 32 F.
5.If the apple pie comes to a temperature of 100 F in
20 minutes, find the value of the constant b in the
above equation. Express your answer to 5 decimal
places.
6.How long will it take for the pie to reach its final
temperature of 200 F?
7. The pie is taken out of the oven and set on a table
in a room at 70 F. In 10 minutes, it has cooled to
180 F. However, the pie must cool to 120 F
before it is ready to eat. How much longer will
you have to wait?