- Topic
- Related rates
- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 2
Related rates practice question 7
A woman 1.5 m tall walks away from a 6 m lamppost along level ground. Let be her distance from the foot of the post and the length of her shadow, both changing with time. At one moment she is 12 m from the post. Which equation holds at every instant?
The lamp, the top of her head and the tip of her shadow lie on one straight line. Which right triangles does that line make with the ground, and what is the base of each?
Two right triangles share the tip of the shadow: a small one for the woman and a large one for the post. Being similar, they have the same height over base.
Theory UBC Math 100 §5CLP-1 §3.2Key results
Worked solution
Strategy
The ray from the lamp that grazes her head ends at the tip of the shadow, so the woman and the post stand in two similar right triangles that share that tip. Match height to base in each, then simplify.
Working
Both triangles have a right angle on the ground and share the angle at the tip of the shadow.
Similar triangles: height over base is the same in both.
Cross-multiply. The shadow is always a third of her distance from the post; at x = 12 it is 4 m.
Answer
The answer is 1.5 over s equals 6 over the quantity x plus s, which simplifies to s equals x over 3.
Check
At x = 12 it gives s = 4: the ray from the lamp falls 4.5 m in 12 m to her head, then 1.5 m in 4 m, the same slope.
Where this goes wrong
Using x as the base of the lamppost’s triangle. The large triangle runs from the post all the way to the tip of the shadow, so its base is x + s.
Leans on
Similar triangles, from Functions, graphs and inverses
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