The exercises below give you the opportunity to take scenarios and turn them into differential equations that model the scenario. Use this opportunity to practice that particular skill.
If you jump right away to thinking about the solution, you will miss out on learning this most important skill - thinking about how things change and putting that into an equation (differential equation).
1.
I make a pot of hot coffee and take it outside to cool. The coffee’s temperature is initially \(100\) degrees Celsius and the outside temperature is \(10\) degrees C. After \(5\) minutes, my temperature gauge says the coffee is \(85\) degrees C. If I want my coffee to cool to a delightful \(40\) degrees C, how much more time do I need to leave it outside?
2.
Now, instead of taking the pot of coffee outside, I put it in my freezer at \(-5\) degrees Celsius. Assume the same constant of proportionality as in the problem above. How long until the coffee is \(40\) degrees now?
3.
My sample of Pylorium Alabastericum (a type of bacteria) doubles in weight every ten hours. Assume the sample is small enough so there’s enough food and space for the colony. How long will it take to grow from \(1\) gram to \(3.5\) grams?
4.
Carbon-14 is used to date some organic artifacts because once the parent material dies, it stops taking in Carbon-14 and this slightly radioactive material begins to decay. Carbon-14 has a half-life of approximately \(5700\) years (that’s how long it takes half of the Carbon-14 to become a more stable isotope of Carbon). If a fragment of jawbone contains \(1.5\%\) of the expected amount of Carbon-14, how long ago did the animal live?
5.
I find a library book that my great-great-grandmother borrowed from the economics library and forgot to return. On the library tag, it tells me that the fine will be \(10\) cents and that the fine will grow exponentially at a \(10\%\) annual rate, compounded continuously. If the book is returned now, \(120\) years later, what fine can I expect to be charged?
6.
I put my roast in the oven. The roast is initially at \(70\) degrees Fahrenheit room temperature and I put in a \(300\) degree oven. After half an hour, I see that the roast is heated to an internal temperature of \(100\) degrees. How long will it take to reach \(165\) degrees Fahrenheit?
7.
Now I take my roast out and let it rest. If it sits for forty minutes while I make the vegetables and salad, what will the internal temperature be when I serve it? Assume the same constant as you found in the exercise above.
8.
I set up a college fund for my daughter when she’s born. I put \(\$100,000\) in it as an initial investment, invested at a stable \(3\%\) annual interest (compounded continuously). Additionally, I add \(\$10,000\) each year on her birthday. After \(18\) years, how much is in the account?
9.
I want to make a batch of nail polish remover containing \(40\%\) acetone. I have a vat of \(1000\) liters of weak acetone in solution, currently at \(10\%.\) I pump in a strong solvent mixture with \(75\%\) acetone, at a rate of \(10\) liters per minute, and let the well-mixed solution drain out at the same rate. As soon as the acetone in my vat reaches a \(40\%\) concentration, I stop this process. What sized waste container do I need to hold the quantity of solution that drains out?
10.
One winter morning, snow begins to fall in my rural community at a constant rate. I wake up and get in my plow to clear my road, setting off at \(8\)am. After one hour, I have cleared \(2\) miles of road. It takes me two more hours to clear the next two miles as the snow keeps falling. Assume that my plow clears snow at a constant rate of volume over time. If \(x(t)\) is the distance my plow is from the start at time \(t\text{,}\) then \(\displaystyle x’(t) = \frac{k}{t}\) for some constant \(k.\) What time did it start snowing?
11.
The next week, it starts snowing again at possibly a different rate from last week. Once again I set off at \(8\) am, traveled \(4\) miles by \(9\) am, and an additional \(3\) miles an hour later. What time did it start to snow this week?
