Test questions: Modeling physical phenomena with first-order differential equations

📚 Quick Study Guide

• 🌡️ Newton's Law of Cooling: The rate of change of an object's temperature is proportional to the difference between its own temperature and the ambient temperature (i.e. the temperature of its surroundings). Mathematically, this is represented as $\frac{dT}{dt} = k(T - T_a)$, where $T$ is the temperature of the object at time $t$, $T_a$ is the ambient temperature, and $k$ is a constant.
• 💰 Mixing Problems: These involve determining the amount of a substance in a tank at a given time. The differential equation is often in the form $\frac{dA}{dt} = (Rate\,in) - (Rate\,out)$, where $A(t)$ is the amount of the substance at time $t$.
• 🦠 Population Growth: If we assume that the rate of growth of a population is proportional to the size of the population, we get the differential equation $\frac{dP}{dt} = kP$, where $P(t)$ is the population size at time $t$, and $k$ is the growth rate.
• 🔌 RL Circuits: In an RL circuit (a circuit with a resistor and an inductor), the current $I$ satisfies the differential equation $L\frac{dI}{dt} + RI = V(t)$, where $L$ is the inductance, $R$ is the resistance, and $V(t)$ is the voltage source.
• ☢️ Radioactive Decay: The rate of decay of a radioactive substance is proportional to the amount of the substance present. This is modeled by $\frac{dN}{dt} = -λN$, where $N(t)$ is the amount of the substance at time $t$, and $λ$ is the decay constant.

🧪 Practice Quiz

1. Question 1: A cup of coffee is initially at 90°C and is left in a room with an ambient temperature of 20°C. After 10 minutes, the coffee is at 60°C. What is the temperature of the coffee after 20 minutes?
   1. 30°C
   2. 40°C
   3. 45°C
   4. 50°C
2. Question 2: A tank contains 1000 liters of water with 10 kg of salt dissolved in it. Water containing 0.01 kg of salt per liter is pumped into the tank at a rate of 5 liters per minute, and the mixture is pumped out at the same rate. What is the amount of salt in the tank after a very long time?
   1. 5 kg
   2. 10 kg
   3. 15 kg
   4. 20 kg
3. Question 3: The population of a certain bacteria grows at a rate proportional to its size. Initially, there are 100 bacteria. After 2 hours, there are 300 bacteria. How many bacteria will there be after 4 hours?
   1. 600
   2. 900
   3. 1200
   4. 2700
4. Question 4: In an RL circuit, $L = 1$ henry, $R = 10$ ohms, and $V(t) = 100$ volts. If the initial current is zero, what is the current as $t$ approaches infinity?
   1. 5 A
   2. 10 A
   3. 15 A
   4. 20 A
5. Question 5: A radioactive substance has a half-life of 10 years. If we start with 100 grams of the substance, how much will remain after 30 years?
   1. 12.5 grams
   2. 25 grams
   3. 50 grams
   4. 75 grams
6. Question 6: A tank initially contains 200 gallons of pure water. Water containing 2 pounds of salt per gallon enters the tank at a rate of 3 gallons per minute, and the well-stirred mixture leaves at the same rate. What is the amount of salt in the tank after 10 minutes?
   1. Approximately 50 lbs
   2. Approximately 55 lbs
   3. Approximately 60 lbs
   4. Approximately 65 lbs
7. Question 7: The temperature of a metal rod at one end is kept at 100°C, and the ambient temperature is 20°C. After a long time, what will the temperature distribution look like along the rod, assuming heat loss is proportional to temperature difference?
   1. Uniformly 100°C
   2. Linearly decreasing from 100°C to 20°C
   3. Exponentially decreasing from 100°C to 20°C
   4. Uniformly 20°C