Here is a new twist on the question. Suppose we want to cool our hot coffee to 100°F as quicklyas possible (sooner than 10 minutes). Suppose also that we have 1 ounce of cold milk with atemperature of 40°F that we can add to the 8 ounces of your coffee at any time. We will try toanswer 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 effectof cold milk on the temperature of the coffee.A reasonable assumption is that when milk isadded to coffee, the temperature of the coffeeimmediately decreases to the average of thecoffee temperature and the milk temperature,where the average is weighted by the volumes,illustrated in the figure on the right.Milk is added-150New temperature of coffee(10, 100)400-50As you can see on the graph, adding the milkimmediately changes the coffee's temperature.So if we add 1 ounce of milk with temperatureTm to 8 ounces of coffee with temperature T,the temperature of the mixture will bet (time in minutes)1. Tm + 8 · TTm + 8TTnew =(2)1+8If the milk is added when the temperature of the coffee is 150 °F as shown in the graph, whatwill be the temperature of the mixture? Does it agree with the graph?AgreesI. 40 + 8.15040 + 1200I new == 137.8°F%3D99. If we wait too long to add the milk, the temperature might fall below 100°F. If we add themilk too soon, the coffee will still need to cool. But there is a point that is "just right" - whereTnew = 100. Solve for the temperature T of the coffee in Equation (2) where this occurs.2408.T240 + 81100 =I +8て=107.5° F3/510. 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 answert. 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)e107.5 =s= 0.125t =8.1 MinWe know that adding milk at any point after t; will make the coffee's temperature less than100°F. What if we add the milk before t;? How much longer will it need to cool? Let'sinvestigate!11. Assume the coffee is allowed to cool for t,milk minutes (after which we will add the milk), wheretmilk will be determined later. Use your solution from 3. and the value of k found in 6. and7. to write down the temperature of the coffee at t = tmilk; call it T*. (Your answer will havetmilk in it.)-0.125tmilnT= 12 + 98 eifo - nそmiit >1.826 Min12. Use Equation (2) to determine the temperature of the coffee when 1 ounce of 40°F milk isadded to 8 ounces of coffee, which has a temperature of T*. Call this new coffee temperatureTnew and write its formula below. Your answer should have tmilk in it.Tnew|• 40 + 8 ·150こ- 137.8°F13. 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)elo2.04° FEmillo%3D* (temperature in F) 14. Use your formula for Tnew from 12. to write a final formula for Tmir (t). Your answer shouldhave t and tmilk in it. (You can use your value of k you found previously.)

Given that:Tnew=137.8°F (as found in 12)Value of k=0.125

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)