- Topic
- Related rates
- Difficulty
- Standard
- Suggested time
- 5 min
- Marks
- 3
Related rates practice question 10
A 1.8 m tall person walks away from the base of a 5.4 m lamppost at 1.5 m/s along level ground. How fast is the tip of their shadow moving away from the lamppost?
The question asks how fast the tip of the shadow moves, not how fast the shadow lengthens. How far is the tip from the post, in terms of the person’s distance and the shadow’s length?
Let be the distance from the post to the person and the length of the shadow. The tip of the shadow is at distance from the post.
Two similar triangles share the tip of the shadow, one formed by the lamppost and one by the person. Height over base is the same in both: .
A rate in m/s, exact or to two decimal places.
Theory UBC Math 100 §5CLP-1 §3.2Key results
Worked solution
Strategy
Use similar triangles to relate the shadow length to the walking distance, express the position of the shadow tip in terms of one variable, and differentiate in time.
Working
The person and the lamppost form similar right triangles with the same shadow tip.
Cross-multiply and solve. The shadow is always half the walking distance.
T is the distance from the post to the tip of the shadow, now in one variable.
Differentiate and substitute. Note the answer does not depend on where the person is.
Answer
The answer is 2.25 m/s.
Check
Starting at the post, the person is 3 m out after 2 s, and the ray from the 5.4 m lamp over their head then lands 4.5 m out: 4.5 m in 2 s.
Where this goes wrong
Answering 0.75 m/s, which is the rate the shadow lengthens (ds/dt), not the rate the tip moves. The tip moves with the person and the lengthening shadow together, so the two rates add.
Leans on
Similar triangles, from Functions, graphs and inverses
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