Here is a new twist on the question. Suppose we want to cool our hot coffee to 100 °F as quickly as possible (sooner than 10 minutes). Suppose also that we have 1 ounce of cold milk with a temperature of 40°F that we can add to the 8 ounces of your coffee at any time. We will try to answer the question: When should we add the milk to cool the coffee to 100°F as quickly as possible? 8. We need to make an assumption about the effect of cold milk on the temperature of the coffee. A reasonable assumption is that when milk is added to coffee, the temperature of the coffee immediately decreases to the average of the coffee temperature and the milk temperature, where the average is weighted by the volumes, illustrated in the figure on the right. Milk is added -150 New temperature of coffee (10, 100) T (temperature in F)
Here is a new twist on the question. Suppose we want to cool our hot coffee to 100 °F as quickly as possible (sooner than 10 minutes). Suppose also that we have 1 ounce of cold milk with a temperature of 40°F that we can add to the 8 ounces of your coffee at any time. We will try to answer the question: When should we add the milk to cool the coffee to 100°F as quickly as possible? 8. We need to make an assumption about the effect of cold milk on the temperature of the coffee. A reasonable assumption is that when milk is added to coffee, the temperature of the coffee immediately decreases to the average of the coffee temperature and the milk temperature, where the average is weighted by the volumes, illustrated in the figure on the right. Milk is added -150 New temperature of coffee (10, 100) T (temperature in F)
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Transcribed Image Text:Here is a new twist on the question. Suppose we want to cool our hot coffee to 100°F as quickly
as possible (sooner than 10 minutes). Suppose also that we have 1 ounce of cold milk with a
temperature of 40°F that we can add to the 8 ounces of your coffee at any time. We will try to
answer the question: When should we add the milk to cool the coffee to 100°F as quickly as possible?
8. We need to make an assumption about the effect
of cold milk on the temperature of the coffee.
A reasonable assumption is that when milk is
added to coffee, the temperature of the coffee
immediately decreases to the average of the
coffee temperature and the milk temperature,
where the average is weighted by the volumes,
illustrated in the figure on the right.
Milk is added
-150
New temperature of coffee
(10, 100)
400
-50
As you can see on the graph, adding the milk
immediately changes the coffee's temperature.
So if we add 1 ounce of milk with temperature
Tm to 8 ounces of coffee with temperature T,
the temperature of the mixture will be
t (time in minutes)
1. Tm + 8 · T
Tm + 8T
Tnew =
(2)
1+8
If the milk is added when the temperature of the coffee is 150 °F as shown in the graph, what
will be the temperature of the mixture? Does it agree with the graph?
Agrees
I. 40 + 8.150
40 + 1200
I new =
= 137.8°F
%3D
9
9. If we wait too long to add the milk, the temperature might fall below 100°F. If we add the
milk too soon, the coffee will still need to cool. But there is a point that is "just right" - where
Tnew = 100. Solve for the temperature T of the coffee in Equation (2) where this occurs.
240
8.T
240 + 81
100 =
I +8
て=
107.5° F
3/5
10. Using your temperature T from 9. with your solution from 3. and the value of k found in 6.
and 7., determine the time needed for the coffee to cool to this temperature. Call your answer
t. By adding milk at that time, how much sooner will we be able to drink our coffee?
Tニ 101.S
-0.125 (t)
72 + (170-72)e
107.5 =
s= 0.125
t =
8.1 Min
We know that adding milk at any point after t; will make the coffee's temperature less than
100°F. What if we add the milk before t;? How much longer will it need to cool? Let's
investigate!
11. Assume the coffee is allowed to cool for t,milk minutes (after which we will add the milk), where
tmilk will be determined later. Use your solution from 3. and the value of k found in 6. and
7. to write down the temperature of the coffee at t = tmilk; call it T*. (Your answer will have
tmilk in it.)
-0.125tmiln
T= 12 + 98 e
ifo - n
そmiit >
1.826 Min
12. Use Equation (2) to determine the temperature of the coffee when 1 ounce of 40°F milk is
added to 8 ounces of coffee, which has a temperature of T*. Call this new coffee temperature
Tnew and write its formula below. Your answer should have tmilk in it.
Tnew
|• 40 + 8 ·150
こ
- 137.8°F
13. We now have that the temperature of the coffee for t > tmilk using Tnew as the initial tempera-
ture at time tmilk is Tmiz (t) = 72+(Tnew – 72)e¬k(t-tmilk). Verify first that Tmiz(tmilk) = Tnew-
- o. 123 (8.1 - 1.826)
I mix (tain)= 72 + "
(137.8-72)e
lo2.04° F
Emillo
%3D
* (temperature in F)
Transcribed Image Text:14. Use your formula for Tnew from 12. to write a final formula for Tmir (t). Your answer should
have t and tmilk in it. (You can use your value of k you found previously.)
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