Sandra's canoeing rate in still water is 6 miles per hour. It takes Sandra 3 hours to travel 10 miles round trip. Assuming a constant rate of speed, determine the rate of the current.
Need steps of solving
[5 / (6 + c)] + [5 / (6 - c)] = 3
multiplying by [(6 + c)(6 - c)] ...30 - 5 c + 30 + 5 c = 108 - 3 c^2
20 = 36 - c^2 ... c^2 = 16
To determine the rate of the current, we can use the formula: speed in still water = speed of the current + speed against the current.
Let's say the speed of the current is represented by "c".
Given:
Speed in still water = 6 miles per hour
Time taken to travel 10 miles round trip = 3 hours
Step 1: Determine Sandra's speed against the current
Since she is traveling round trip, the distance she travels against the current is half the total distance. Therefore, she travels (10 miles / 2) = 5 miles against the current.
Step 2: Determine Sandra's speed with the current
To determine the time taken to travel with the current, we subtract the time taken to travel against the current from the total time.
Time taken to travel against the current = 3 hours
Time taken to travel with the current = total time - time taken against the current
Time taken to travel with the current = 3 hours - 3 hours = 0 hours
Step 3: Use the formula to determine the rate of the current:
speed in still water = speed of the current + speed against the current
Plugging in the values:
6 miles per hour = c + 5 miles / 3 hours
Step 4: Solve for the rate of the current (c):
6 miles per hour = c + 5 miles / 3 hours
Multiplying both sides by 3:
18 miles per hour = 3c + 5 miles
Subtracting 5 miles from both sides:
13 miles per hour = 3c
Dividing both sides by 3:
Rate of the current (c) = 13 miles per hour / 3
c ≈ 4.33 miles per hour
Therefore, the rate of the current is approximately 4.33 miles per hour.
To determine the rate of the current, follow these steps:
1. Let's assume the rate of the current is represented by 'c' (in miles per hour).
2. The total distance of the round trip is 10 miles, so Sandra goes 5 miles one way and 5 miles back.
3. Let's represent Sandra's speed relative to the current as x miles per hour when going downstream (in the direction of the current). In this case, her speed when going upstream (against the current) would be (6 - x) miles per hour.
4. When going downstream, the current is assisting her, so her effective speed is (6 + c) miles per hour. The time it takes for Sandra to travel the downstream distance of 5 miles is 5/(6 + c) hours.
5. When going upstream, the current is opposing her, so her effective speed is (6 - c) miles per hour. The time it takes for Sandra to travel the upstream distance of 5 miles is 5/(6 - c) hours.
6. According to the information given, it takes Sandra a total of 3 hours to complete the round trip. So the sum of the downstream and upstream times is 3 hours:
5/(6 + c) + 5/(6 - c) = 3
7. Now, you can solve this equation to find the value of 'c', the rate of the current. Simplify and solve for 'c':
Multiply every term by (6 + c)(6 - c) to remove the denominators:
5(6 - c) + 5(6 + c) = 3(6 + c)(6 - c)
30 - 5c + 30 + 5c = 3(36 - c^2)
60 = 108 - 3c^2
Rearrange and simplify:
3c^2 = 48
c^2 = 16
c = ±4
8. Since the current cannot have a negative speed, the rate of the current is 4 miles per hour.
Therefore, the rate of the current is 4 miles per hour.