Verified Answer :
- Establish the speed equations for upstream (\(v_c - v_w = 8\)) and downstream (\(v_c + v_w = 16\)) trips.
- Use the elimination method to solve the system of linear equations by adding the two equations: \(2v_c = 24\).
- Find the canoe's speed in still water: \(v_c = 12 \text{ km/h}\).
- Find the current speed by substitution: \(v_w = 4 \text{ km/h}\).
- The final speeds are \(\boxed{12 \text{ km/h}}\) and \(\boxed{4 \text{ km/h}}\).
Explanation
- Data and Problem Analysis
We are given that a canoe travels a distance of \(40 \text{ km}\) upstream in \(5 \text{ hours}\). On the downstream trip, it travels the same distance of \(40 \text{ km}\) in half the time, which is \(2.5 \text{ hours}\). Let \(v_c\) be the speed of the canoe in still water and \(v_w\) be the speed of the water current. When moving upstream, the canoe's speed against the current is \(v_c - v_w\). When moving downstream, the canoe's speed with the current is \(v_c + v_w\). The relationship between distance, speed, and time is given by the formula \(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\).
- Setting up the Equations
Using the formula \(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\), we can set up the equations for both trips:
- Upstream trip: \(v_c - v_w = \frac{40 \text{ km}}{5 \text{ h}} = 8 \text{ km/h}\)
- Downstream trip: \(v_c + v_w = \frac{40 \text{ km}}{2.5 \text{ h}} = 16 \text{ km/h}\)
This gives us a system of two linear equations with two variables:
1) \(v_c - v_w = 8\)
2) \(v_c + v_w = 16\)
- Solving the System of Equations
To solve the system of equations, we can add the two equations together:
\((v_c - v_w) + (v_c + v_w) = 8 + 16\)
\(2v_c = 24\)
Dividing by \(2\), we find the speed of the canoe in still water:
\(v_c = 12 \text{ km/h}\)
Now, substitute \(v_c = 12\) into the second equation:
\(12 + v_w = 16\)
Subtracting \(12\) from both sides gives the speed of the current:
\(v_w = 4 \text{ km/h}\)
- Conclusion
The speed of the canoe in still water is \(12 \text{ km/h}\) and the speed of the current is \(4 \text{ km/h}\). Thus, the rate of the canoe in still water is \(\boxed{12 \text{ km/h}}\) and the rate of the current is \(\boxed{4 \text{ km/h}}\).
Examples
Understanding the relationship between speed, current, and travel time is crucial for navigation and planning. For example, if you are planning a rowing or kayaking trip, knowing how the river current affects your speed allows you to estimate travel times accurately and plan your stops effectively. This ensures you can complete your journey safely and return in time.