Why Ladder and Shadow Problems Matter
Ladder and shadow problems are classic AP Calculus related-rates questions because several quantities change at the same time. A ladder slides down a wall, a person's shadow changes length, or a light source creates a moving shadow. The calculus itself is usually straightforward. The difficult part is building the correct geometric equation.
The central idea is simple: find an equation connecting the changing quantities, differentiate that equation with respect to time, and then substitute the information from the specific instant.
Do not start by plugging in the numerical values. First write the geometric relationship using variables. Then differentiate. Only after the derivative equation is established should you substitute the given values.
The Related-Rates Framework
Suppose two or more quantities depend on time. For example, let represent a ladder's horizontal distance from a wall and represent its vertical height on the wall. If the ladder has fixed length , then the Pythagorean theorem gives a relationship between them.
Differentiating with respect to time gives:
Notice what happened: the equation now contains the rates and . Those are the quantities that related-rates questions usually ask you to find.
- Draw and label the geometry.
- Choose variables for the changing quantities.
- Write one equation connecting those variables.
- Differentiate with respect to time.
- Substitute the values from the requested instant.
- Solve for the unknown rate.
- Interpret the sign and include units.
Ladder Problems
A ladder leaning against a vertical wall forms a right triangle. If the ladder's length stays constant, then the horizontal distance from the wall and the vertical height satisfy the Pythagorean theorem.
Let:
- = horizontal distance from the wall
- = height of the top of the ladder on the wall
- = fixed ladder length
Because is constant, its derivative is zero. Differentiate implicitly with respect to time:
If the bottom of the ladder moves away from the wall, then is positive. The equation therefore forces to be negative: the top of the ladder must be sliding downward.
Worked Ladder Example
A 10-foot ladder leans against a wall. The bottom of the ladder is moving away from the wall at 2 ft/s. How fast is the top of the ladder moving downward when the bottom is 6 feet from the wall?
Why the Answer Is Negative
In related rates, a negative derivative usually does not mean the calculation is wrong. It tells you the direction of change. Here, the top of the ladder's height is decreasing, so its rate of change is negative.
If the question asks for the rate at which the height is changing, the signed answer is negative. If it asks how fast the top is moving downward, report the positive speed together with the word "downward."
A Ladder Problem with a Different Unknown
A 13-foot ladder leans against a wall. The top is 12 feet above the ground and is sliding downward at 3 ft/s. How fast is the bottom of the ladder moving away from the wall?
Shadow Problems
Shadow problems are another application of related rates. The key observation is that the triangles formed by the object and its shadow are often similar triangles.
Instead of using the Pythagorean theorem, you usually write a proportion relating corresponding sides of the two triangles.
When a light source creates a shadow, identify the large triangle from the light source to the top of the object and the end of the shadow. Then identify the smaller triangle formed by the light source, the object, and the ground. Their corresponding sides give the equation you need.
Basic Shadow Relationship
Suppose a light is positioned a fixed distance from a vertical object. Let be the object's height and be the length of its shadow. If the light is from the object, similar triangles produce a relationship of the form:
Solving for the shadow length gives:
This is exactly what we want in a related-rates problem: an equation involving the quantities that are changing.
Worked Shadow Example
A 12-foot lamppost stands on level ground. A person who is 6 feet tall walks directly away from the lamppost at 4 ft/s. How fast is the person's shadow increasing in length?
The Important Trick in Shadow Problems
The most common source of confusion is deciding what the variable actually represents. In many shadow problems, the distance from the light source to the person is not the same thing as the length of the person's shadow.
If the person is from the lamp and the shadow has length , then the base of the large triangle is , not simply .
That distinction is often the entire challenge of the problem.
Ladder vs. Shadow Problems
Common Mistakes
The variables need to remain variables while you differentiate. If you replace them with their numerical values too early, you can destroy the relationship between the changing quantities.
In a typical lamppost problem, the large triangle's base includes both the person's distance from the lamp and the person's shadow. Carefully label the diagram before writing the proportion.
If the ladder's length is constant, then is constant, so its time derivative is zero.
A negative rate often communicates direction. A ladder's height may be decreasing, while its horizontal distance is increasing. The signs should agree with the physical situation.
Related rates is not primarily a differentiation problem. It is a modeling problem followed by differentiation. If the equation is wrong, perfect differentiation will still produce the wrong answer.
A Fast AP Exam Checklist
- Draw the picture.
- Label every changing quantity.
- Identify the quantity the question asks for.
- Find the geometric relationship.
- Differentiate with respect to time.
- Substitute values only after differentiating.
- Check whether the sign makes physical sense.
- State the answer with units.
Practice Problems
A 15-foot ladder leans against a wall. The bottom moves away from the wall at 3 ft/s. How fast is the top moving downward when the bottom is 9 feet from the wall?
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A 25-foot ladder leans against a wall. The top is 24 feet above the ground and slides downward at 5 ft/s. How fast is the bottom moving away from the wall?
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A 10-foot pole casts a shadow. A person walks directly away from a point light source toward the end of the shadow. Set up a similar-triangle equation and explain why the distance from the light to the shadow's endpoint is not simply the shadow length.
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A 15-foot lamppost illuminates a 5-foot-tall person. The person walks away from the lamppost at 3 ft/s. How fast is the person's shadow increasing in length?
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A 20-foot ladder leans against a wall. The bottom is moving away from the wall at 4 ft/s. Find the rate at which the top is descending when the top is 12 feet above the ground.
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What to Remember
Ladder and shadow problems become much easier when you stop thinking of them as mysterious "word problems" and instead treat them as geometry that happens to change over time.
For ladders, the Pythagorean theorem is usually the starting point. For shadows, similar triangles are usually the key. In both cases, the same related-rates process applies:
The derivative is only the middle of the problem. The most important step is building the correct relationship between the quantities.
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