A motorboat, whose speed is 18 km/hr in still water, takes one hour more to go…
2023
A motorboat, whose speed is 18 km/hr in still water, takes one hour more to go 24 km upstream than to return downstream to the same spot. The speed of the stream is
Answer: B. 6 km/hr — ConceptLet b be the boat speed in still water and v the stream speed. Then the effective upstream speed is b - v and the downstream speed is b + v. For…
- A.
−54 km/hr
- B.
6 km/hr
- C.
10 km/hr
- D.
More than one of the above
- E.
None of the above
Attempted by 16 students.
Show answer & explanation
Correct answer: B
Concept
Let b be the boat speed in still water and v the stream speed. Then the effective upstream speed is b - v and the downstream speed is b + v.
For distance d, travel time is d divided by speed. Therefore, the excess upstream time is d/(b - v) - d/(b + v). A physical stream speed must satisfy 0 ≤ v < b.
Application
Here b = 18 km/h, d = 24 km, and the upstream trip takes 1 hour longer, so 24/(18 - v) - 24/(18 + v) = 1.
Combining the fractions gives 24[(18 + v) - (18 - v)]/[(18 - v)(18 + v)] = 1, hence 48v/(324 - v2) = 1.
Thus 48v = 324 - v2, so v2 + 48v - 324 = 0.
Factoring gives (v - 6)(v + 54) = 0, so the algebraic roots are v = 6 and v = -54.
Because a stream speed is a magnitude and must satisfy 0 ≤ v < 18, v = -54 is inadmissible. Hence the stream speed is 6 km/h.
Cross-check
At v = 6, the upstream speed is 18 - 6 = 12 km/h and the downstream speed is 18 + 6 = 24 km/h. The corresponding times are 24/12 = 2 hours and 24/24 = 1 hour, whose difference is exactly 1 hour.
Result: The speed of the stream is 6 km/h.