- Topic
- Optimization
- Difficulty
- Capstone
- Suggested time
- 10 min
- Marks
- 5
Optimization practice question 17
A rowing boat is 6 km offshore from the nearest point on a straight coastline. The rower wants to reach a camp on the coast 10 km from . She can row at 5 km/h and walk along the coast at 13 km/h. To minimise her total travel time, how far from should she land?
Give the distance from A in kilometres.
Call the landing point km from . Each leg takes distance over speed: how far does she row, how far does she walk, and what values can take?
Let her land km from . Rowing distance ; walking distance . Total time for .
Set : . Square carefully, and check the endpoints too.
A number, as a fraction or a decimal.
Theory UBC Math 100 §8CLP-1 §3.5Key results
Worked solution
Strategy
Model the time as a function of the landing point, differentiate, solve for the critical point, and compare against the endpoints because the domain is a closed interval.
Working
Time is distance over speed for each leg.
Chain rule on the square root.
Square both sides (x > 0, so no extraneous root).
The critical point beats both endpoints: land 2.5 km from A, in hours about 1.88.
Answer
The answer is 2.5 kilometres from A.
Check
Landing 2 km or 3 km from A takes about 112.8 minutes, against 112.6 minutes for 2.5 km.
Where this goes wrong
Squaring 13x = 5√(36 + x²) as 169x² = 25·36 + x², dropping the 25 on the x² term. Keep the 25 on the whole bracket; and always compare with the endpoints, since rowing straight to A or straight to the camp are real options.
Leans on
The chain rule on a square root, from The chain rule and logarithms
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