How to Solve a Related Rate Problem

Related rate problems are a type of calculus problem that involves finding the rate of change of one variable with respect to another variable, when both variables are changing at the same time. They are often used to solve real-world problems, such as finding the rate at which a ladder slides down a wall, or the rate at which the area of a circle is changing.

To solve a related rate problem, follow these steps:

1. **Identify the given information.** This will include the rates of change of the known variables, as well as the relationship between the variables.
2. **Write an equation that relates the variables.** This equation will often be derived from a geometric or physical relationship.
3. **Differentiate the equation with respect to time.** This will give you an equation that relates the rates of change of the variables.
4. **Substitute the given information into the equation.** This will give you an equation that you can solve for the unknown rate of change.

Here is an example of a related rate problem:

> A ladder 10 feet long is leaning against a wall. The bottom of the ladder is sliding away from the wall at a rate of 2 feet per second. How fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 feet from the wall?

To solve this problem, we can use the following steps:

1. **Identify the given information.**
– The rate of change of the bottom of the ladder is -2 feet per second.
– The length of the ladder is 10 feet.
– The bottom of the ladder is 6 feet from the wall.
2. **Write an equation that relates the variables.**
– Using the Pythagorean theorem, we can write the following equation:
“`
a^2 + b^2 = c^2
“`
– where:
– a is the distance from the bottom of the ladder to the wall
– b is the distance from the top of the ladder to the ground
– c is the length of the ladder
3. **Differentiate the equation with respect to time.**
– Differentiating both sides of the equation with respect to time, we get:
“`
2a * da/dt + 2b * db/dt = 0
“`
– where:
– da/dt is the rate of change of the distance from the bottom of the ladder to the wall
– db/dt is the rate of change of the distance from the top of the ladder to the ground
4. **Substitute the given information into the equation.**
– We are given that da/dt = -2 feet per second and a = 6 feet. Substituting these values into the equation, we get:
“`
2 * 6 * (-2) + 2b * db/dt = 0
“`
– Simplifying, we get:
“`
-24 + 2b * db/dt = 0
“`
– Solving for db/dt, we get:
“`
db/dt = 12 / b
“`
5. **Evaluate db/dt at b = 8 feet.**
– When the bottom of the ladder is 6 feet from the wall, the distance from the top of the ladder to the ground is 8 feet. Substituting this value into the equation for db/dt, we get:
“`
db/dt = 12 / 8 = 1.5 feet per second
“`

Therefore, the top of the ladder is sliding down the wall at a rate of 1.5 feet per second when the bottom of the ladder is 6 feet from the wall.

Here are some additional tips for solving related rate problems:

– Draw a diagram to help you visualize the problem.
– Be sure to identify all of the variables involved, as well as their rates of change.
– Write an equation that relates the variables.
– Differentiate the equation with respect to time.
– Substitute the given information into the equation.
– Solve for the unknown rate of change.

With practice, you will be able to solve related rate problems quickly and easily.

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