§7.5 Modeling with differential equations (continued)

Summary

Congratulations, you just won the lottery! In one option presented to you, you will be paid one million dollars a year for the next 25 years. You can deposit this money in an account that will earn 5% each year. - Set up a differential equation that describes the rate of change in the amount of money in the account. Two factors cause the amount to grow——first, you are depositing one millon dollars per year and second, you are earning 5% interest. - If there is no amount of money in the account when you open it, how much money will you have in the account after 25 years? - The second option presented to you is to take a lump sum of 10 million dollars, which you will deposit into a similar account. How much money will you have in that account after 25 years? - Do you prefer the first or second option? Explain your thinking. - At what time does the amount of money in the account under the first option overtake the amount of money in the account under the second option?

When a skydiver jumps from a plane, gravity causes her downward velocity to increase at the rate of $g\approx 9.8$ meters per second squared. At the same time, wind resistance causes her velocity to decrease at a rate proportional to the velocity. - Using $k$ to represent the constant of proportionality, write a differential equation that describes the rate of change of the skydiver's velocity. - Find any equilibrium solutions and decide whether they are stable or unstable. Your result should depend on $k$. - Suppose that the initial velocity is zero. Find the velocity $v(t)$. - A typical terminal velocity for a skydiver falling face down is 54 meters per second. What is the value of $k$ for this skydiver? - How long does it take to reach 50% of the terminal velocity?

During the first few years of life, the rate at which a baby gains weight is proportional to the reciprocal of its weight. - Express this fact as a differential equation. - Suppose that a baby weighs 8 pounds at birth and 9 pounds one month later. How much will he weigh at one year? - Do you think this is a realistic model for a long time?

Suppose that you have a water tank that holds 100 gallons of water. A briny solution, which contains 20 grams of salt per gallon, enters the tank at the rate of 3 gallons per minute.

At the same time, the solution is well mixed, and water is pumped out of the tank at the rate of 3 gallons per minute. - Since 3 gallons enter the tank every minute and 3 gallons leave every minute, what can you conclude about the volume of water in the tank? - How many grams of salt enter the tank every minute? - Suppose that $S(t)$ denotes the number of grams of salt in the tank in minute $t$. How many grams are there in each gallon in minute $t$? - Since water leaves the tank at 3 gallons per minute, how many grams of salt leave the tank each minute? - Write a differential equation that expresses the total rate of change of $S$. - Identify any equilibrium solutions and determine whether they are stable or unstable. - Suppose that there is initially no salt in the tank. Find the amount of salt $S(t)$ in minute $t$. - What happens to $S(t)$ after a very long time? Explain how you could have predicted this only knowing how much salt there is in each gallon of the briny solution that enters the tank.