More on differential equations and Newton's Law of Cooling.

1. Cooling Colas
or
more about differential equations
Ice cold by flickr user nicholasjon

2. Newton's Law of Cooling ...
According to Newton’s law of cooling, a hot object cools at a rate
proportional to the difference between its own temperature and that of
its environment. If a roast at room temperature, 68°F, is put into a 20°F
freezer, and if, after 2 hours, the temperature of the roast is 40°F,
(a) What is the temperature of the roast after 5 hours?

3. Newton's Law of Cooling ...
According to Newton’s law of cooling, a hot object cools at a rate
proportional to the difference between its own temperature and that of
its environment. If a roast at room temperature, 68°F, is put into a 20°F
freezer, and if, after 2 hours, the temperature of the roast is 40°F,
(a) What is the temperature of the roast after 5 hours?

4. Newton's Law of Cooling ...
According to Newton’s law of cooling, a hot object cools at a rate
proportional to the difference between its own temperature and that of
its environment. If a roast at room temperature, 68°F, is put into a 20°F
freezer, and if, after 2 hours, the temperature of the roast is 40°F,
(b) How long will it take for the temperature of the roast to fall to 21°F?

5. Cooling Colas
A few weekends ago we invited some friends over and served
them lunch. The day before we had bought some cans of soda,
but we didn't have enough room in our refrigerator to cool the
colas. Since it was February my wife suggested that we leave the
cans outside overnight to cool.
The overnight temperature was to be in the twenties, so I was
afraid they might freeze; however, I figured out a way to
approximate the temperature of the cola at any given time.
First, I read the thermometer in our house. It said 72°F. Next I
read our outdoor thermometer, and it read 25°F. I put the cola
outside for 30 minutes and then brought one of the cans inside to
have a drink. Before I drank the soda, I measured the
temperature. The temperature of the soda was 60°F.
Based on this information, how long would it take the cola to cool to
35°F? Assume that the outdoor temperature remains constant during
the cooling process.

6. Cooling Colas
A few weekends ago we invited some friends over and served
them lunch. The day before we had bought some cans of soda,
but we didn't have enough room in our refrigerator to cool the
colas. Since it was February my wife suggested that we leave the
cans outside overnight to cool.
The overnight temperature was to be in the twenties, so I was
afraid they might freeze; however, I figured out a way to
approximate the temperature of the cola at any given time.
First, I read the thermometer in our house. It said 72°F. Next I
read our outdoor thermometer, and it read 25°F. I put the cola
outside for 30 minutes and then brought one of the cans inside to
have a drink. Before I drank the soda, I measured the
temperature. The temperature of the soda was 60°F.
Based on this information, how long would it take the cola to cool to
35°F? Assume that the outdoor temperature remains constant during
the cooling process.

7. First, I read the thermometer in our house. It said 72°F. Next I
read our outdoor thermometer, and it read 25°F. I put the cola
outside for 30 minutes and then brought one of the cans inside to
have a drink. Before I drank the soda, I measured the
temperature. The temperature of the soda was 60°F.

8. First, I read the thermometer in our house. It said 72°F. Next I
read our outdoor thermometer, and it read 25°F. I put the cola
outside for 30 minutes and then brought one of the cans inside to
have a drink. Before I drank the soda, I measured the
temperature. The temperature of the soda was 60°F.

9. Based on this information, how long would it take the cola to cool to
35°F? Assume that the outdoor temperature remains constant during
the cooling process.

10. Hot Cocoa
You have just poured yourself a nice mug of hot chocolate. As
you lift up the mug to take a sip, you realize the hot chocolate is
too hot. You don't want to burn your tongue, so you decide to
wait.
You walk over to the thermometer on the wall and note that the
room temperature is 70°F degrees. You assume that the
temperature of the hot chocolate is about 200°F degrees,
because you added water that was almost boiling to the mug. You
would like to wait until the hot chocolate is about 150°F degrees.
You use a thermometer to measure the temperature of the hot
chocolate. After 2 minutes of cooling, it is at 190°F degrees.
To the nearest minute, how much longer will you need to wait?
Answer on next slide ...

11. It will take 12 minutes after
the coffee begins cooling
from 200 degrees in a 70
degree room to reach 150
degrees. You have already
waited 2 minutes. Since 12 - 2
= 10 there remain 10 minutes
before you can comfortably
take a sip.