Related Rates: Shrinking and Growing Shapes

What Related Rates Problems Are Really Asking

A related rates problem describes a situation in which two or more quantities are changing with time. The goal is usually to determine how quickly one quantity is changing when you know the rate at which another quantity is changing.

The important idea is that the quantities are connected by a mathematical relationship. For geometric problems, that relationship usually comes from a familiar formula such as the area of a circle, the volume of a sphere, or the area of a rectangle.

For example, if a circle is shrinking, its radius and area are both changing. They are not independent: the area is determined by the radius. Once you write that relationship and differentiate it with respect to time, you can connect the two rates.

The Core Related-Rates Formula

Suppose two quantities and are related by an equation . Because both quantities may change with time, differentiate both sides with respect to .

The key is that you are differentiating with respect to time, not with respect to one geometric variable. That means quantities such as are treated as functions of time. Their derivatives become rates such as .

The Four-Step Method

1. Write the geometric relationship. Start with a formula connecting the quantities that are changing.
2. Differentiate with respect to time. Every changing variable gets a derivative with respect to .
3. Substitute the known values and rates. Use the specific instant described in the problem.
4. Solve for the unknown rate. Pay close attention to positive and negative signs and include the correct units.

One of the biggest differences between related rates and ordinary derivative problems is when you substitute values. In most related rates problems, you should differentiate first and substitute the numerical values afterward.

Example 1: A Shrinking Circle

Notice the negative sign. The radius is decreasing, so is negative. Consequently, the area is also decreasing, giving a negative value for . If the question asks "how quickly is the area decreasing?" you can report the positive magnitude and say that the area is decreasing at that rate.

Example 2: A Growing Circle

This example is particularly simple because the circumference is directly proportional to the radius. Whenever the radius grows at a constant rate, the circumference grows at a constant rate as well.

Example 3: A Growing Sphere

The important feature here is that the volume does not increase at the same rate as the radius. Because volume depends on , the volume's rate of change depends strongly on the current radius. A larger balloon can gain volume much faster even if its radius is increasing at the same rate.

Example 4: A Shrinking Rectangle

Both dimensions are shrinking, so both contributions to the area rate are negative. That is exactly what we should expect: if both the length and width are decreasing, the area must also be decreasing.

Example 5: A Growing Circle with Area and Radius

Why You Should Differentiate Before Substituting

This is one of the most important habits in related rates. Suppose you know that a circle currently has radius 5 cm. It can be tempting to immediately replace with 5 in the original equation. But the equation describes the relationship between the quantities while they are changing.

Instead, first differentiate:

Only after you have the rate equation should you plug in the particular radius and known rate. This preserves the relationship between the changing quantities.

Growing vs. Shrinking: Reading the Sign

Related rates problems often contain words such as increasing, decreasing, growing, and shrinking. Translate those words into signs.

• Increasing / growing: the derivative is positive.
• Decreasing / shrinking: the derivative is negative.
• Constant: the derivative is zero.

For example, if a radius is shrinking at 2 cm/min, write , not . The negative sign contains important information: the radius is moving downward as time increases.

A Useful Geometric Formula Sheet

Many shrinking-and-growing related rates problems become much easier if you recognize the correct geometric relationship immediately.

You do not need to memorize a separate related-rates formula for every shape. Instead, start with the familiar geometry formula and differentiate it with respect to time.

Common Mistakes

The Big Picture

Shrinking-and-growing shape problems are really applications of one simple idea: related quantities remain connected even while they change.

The geometry gives you the relationship. Differentiation converts that relationship into a relationship between rates. The numerical values then tell you what is happening at one particular instant.

On an AP Calculus free-response question, the process is usually:

1. Identify the changing quantities.
2. Write the geometric equation connecting them.
3. Differentiate with respect to time.
4. Substitute the given values and rates.
5. Solve for the requested rate.
6. State the answer with units and an appropriate interpretation.

Once you recognize that pattern, a problem about a shrinking circle and a problem about a growing sphere are essentially the same problem wearing different geometric clothing.

Practice Problems