Math Problem Statement
A motorboat whose speed in still water is 18 km/h takes 1 hr more to go 24 km upstream than to return downstream to the same spot. Find the speed of the current
Solution
Let the speed of the current be km/h.
- The speed of the motorboat in still water is 18 km/h.
- The distance to travel upstream and downstream is 24 km.
Upstream:
- The effective speed when going upstream (against the current) is km/h.
- Time taken to travel upstream is given by:
Downstream:
- The effective speed when going downstream (with the current) is km/h.
- Time taken to travel downstream is:
Given Condition:
It is given that it takes 1 hour more to go upstream than downstream. This translates to the equation: Substituting the expressions for the times: Now, let's solve this equation step by step.
Taking the LCM of the left-hand side: Simplifying the numerator: Simplify further: Rearranging the terms: Now, solve this quadratic equation using the quadratic formula: Here, , , and .
So, we have two possible solutions: Since the speed of the current cannot be negative, the speed of the current is:
Conclusion:
The speed of the current is .
Would you like further details or have any questions?
Here are 5 related questions to expand on this:
- How would the problem change if the boat's speed in still water was different?
- What if the distance traveled was 30 km instead of 24 km?
- How would you solve the problem if the time difference was 2 hours instead of 1 hour?
- What is the effect of increasing the current speed on the time taken for upstream travel?
- How does the quadratic formula help in finding the speed of the current?
Tip: When working with speed and current problems, always set up the time equations carefully to reflect the upstream and downstream differences clearly.
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Math Problem Analysis
Mathematical Concepts
Algebra
Time-Distance-Speed Relationships
Quadratic Equations
Formulas
Time = Distance / Speed
Quadratic equation formula: ax^2 + bx + c = 0
Theorems
Quadratic formula
Suitable Grade Level
Grades 9-11
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