From 11:00PM PDT on Friday, July 1 until 5:00AM PDT on Saturday, July 2, the Shmoop engineering elves will be making tweaks and improvements to the site. That means Shmoop will be unavailable for use during that time. Thanks for your patience!
We have changed our privacy policy. In addition, we use cookies on our website for various purposes. By continuing on our website, you consent to our use of cookies. You can learn about our practices by reading our privacy policy.

Word Problems Exercises

Example 1

Translate the English statement into a differential equation. Be sure to specify what your variables are.

The population is increasing at a rate of 1,000 people per year.

Example 2

Translate the English statement into a differential equation. Be sure to specify what your variables are.

The number of bunnies in the forest is increasing at a rate proportional to the number of bunnies there already.

Example 3

Translate the English statement into a differential equation. Be sure to specify what your variables are.

Tamara spends \$40 per week.

Example 4

Translate the English statement into a differential equation. Be sure to specify what your variables are.

Ben receives 20 pieces of junk mail every day.

Example 5

Translate the English statement into a differential equation. Be sure to specify what your variables are.

A batch of cookies is placed in a 375°F oven. The temperature of the cookies increases at a rate proportional to the difference between the temperature of the cookies and the temperature of the oven.

Example 6

The population of bunnies B is increasing at a rate proportional to the size of B. This situation can be modeled with the differential equation

Is the constant k positive or negative?

Example 7

The population of geese G is decreasing at a rate proportional to G. This situation can be modeled with the differential equation

Is the constant k positive or negative?

Example 8

If Q is a positive quantity and

where k < 0, is Q increasing or decreasing?

Example 9

Cookies are placed in a 375°F degree oven to bake. Newton's Law of Heating  says that the temperature t of the cookies will increase at a rate proportional to the difference between the temperature of the surrounding oven and the temperature of the cookies. If we model this situation by the differential equation

is the constant k positive or negative?

Example 10

A hot cup of coffee is placed on the kitchen table in a room that is 68°F. Newton's Law of Cooling says that the temperature t of the coffee will decrease at a rate proportional to the difference between the temperature of the surrounding room and the temperature of the coffee. This situation can be modeled by the differential equation

Is the constant k positive or negative?

Example 11

Model the situation using a differential equation. State the units of each variable and the units of the derivative.

Water is rushing into a tank at a rate of 5 gallons per minute and rushing out again at a rate of 3 gallons per minute.

Example 12

Model the situation using a differential equation. State the units of each variable and the units of the derivative.

The birth rate of the wolf population is 5 percent per year and hunters kill 200 wolves per year.

Example 13

Model the situation using a differential equation. State the units of each variable and the units of the derivative.

Every month Donna's savings account earns 2 percent interest and she deposits an additional \$100 dollars.

Example 14

Model the situation using a differential equation. State the units of each variable and the units of the derivative.

A beach is eroding by 10 percent per year. Every month the waves deposit an extra 150 cubic feet of sand.

Example 15

Model the situation using a differential equation. State the units of each variable and the units of the derivative.

A wasp population has a 10 percent birth rate and a 9 percent death rate. Every year people swat an additional 300 wasps.