Calculus I · Unit 2B · lesson
Shadows and Similar Triangles
Learning objectives
Build and differentiate similar-triangle relations for shadow problems.
Shadows and Proportional Geometry
Before the formulas
Related-rates work in Shadows and Similar Triangles becomes manageable when geometry and calculus are kept separate. Geometry supplies an equation such as the Pythagorean theorem, a volume formula, or a similar-triangle proportion. Calculus differentiates that equation. The derivative does not invent the geometry for you.
Before solving, predict the sign of the unknown rate from the picture. That prediction is a powerful error check. If a ladder foot moves away from a wall, the top should move down; a positive answer for its height rate should trigger a review.
Read this graph as text
Shadow problems are similar-triangle problems. A light, a person, and the shadow tip form one large triangle. The person's height and shadow form a smaller triangle. Their proportional sides connect walking speed to shadow growth. The large triangle has height H and base x+s . The smaller triangle has height h and base s . Similarity gives H/(x+s)=h/s . Notice that the shadow tip's distance from the light is x+s , not just s .
Every relationship in shadow problems are similar-triangle problems is identified with written labels plus distinct solid, dashed, dotted, double, marker, or pattern cues; color is never the only carrier of meaning.
Why it matters: The visual should prevent variable ambiguity. Students frequently use the same symbol for the person's distance from the light and the shadow length, or forget that the shadow tip is farther from the light by their sum.
A light, a person, and the shadow tip form one large triangle. The person's height and shadow form a smaller triangle. Their proportional sides connect walking speed to shadow growth.
Similar triangles convert a moving light picture into an equation
Shadow problems contain two triangles that share an angle and therefore have proportional side lengths. The geometry equation should be built from those proportions before any differentiation occurs.
Define carefully whether a variable measures the person's distance from the light, the shadow length, or the distance from the light to the shadow tip. Confusing those distances is the standard source of a correct derivative applied to the wrong model.
Shadow problems are proportional-geometry problems in motion. Similar triangles relate heights and horizontal distances, but the wording may ask for the shadow length, the tip position, or the rate at which those two separate quantities change.
Label the complete distance from the light to the tip before writing a proportion. Most shadow errors are not calculus errors; they are diagrams that silently changed which segment a symbol represents.
Walking away from a streetlight
A -foot person walks away from a -foot lamp at ft/s. How fast does the tip of the shadow move?
Worked solution
Write a real attempt before opening the supplied answer.
Shadow problems often contain three horizontal distances: lamp to person, person to tip, and lamp to tip. Name them before writing proportions. Otherwise a correct similar-triangle equation is mostly a matter of luck.
After the explanation
Use the section idea
Freeze the geometry at one instant, but differentiate the relationship while every changing quantity is still a function of time.
The picture supplies a constraint; implicit differentiation transmits known rates through that constraint to the unknown rate.
Draw and label first, write one relationship, differentiate with time, then substitute the snapshot measurements and rates.
Substituting numerical dimensions before differentiating and thereby erasing the very change the problem asks about.
Does your final rate have the predicted sign, the correct units, and a magnitude compatible with the diagram?
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