The Connecticut River flows at a rate of 3 km / hour for the length of a popular scenic route. If a cruiser to travels 3 hours with the current to reach a drop-off point, but the return trip against the same current took 7 hours. Find the speed of the boat without a current?
To solve this problem, let's consider the speed of the boat without the current as "x" km/h.
When traveling downstream (with the current), the effective speed of the boat will be the sum of the boat's speed and the speed of the current. Therefore, the effective speed downstream is (x + 3) km/h.
When traveling upstream (against the current), the effective speed of the boat will be the difference between the boat's speed and the speed of the current. Therefore, the effective speed upstream is (x - 3) km/h.
We are given that it takes 3 hours to travel downstream and 7 hours to travel upstream. We can use the formula: time = distance / speed to find the distance traveled in each direction.
For the downstream trip:
Distance = Speed * Time = (x + 3) km/h * 3 hours = 3(x + 3) km.
For the upstream trip:
Distance = Speed * Time = (x - 3) km/h * 7 hours = 7(x - 3) km.
Since the distance traveled in both directions is the same (it's a round trip), we can set the two equations equal to each other:
3(x + 3) = 7(x - 3).
Expanding and simplifying the equation:
3x + 9 = 7x - 21.
Rearranging the equation:
4x = 30.
Dividing both sides by 4:
x = 7.5.
Therefore, the speed of the boat without the current is 7.5 km/h.
To find the speed of the boat without a current, we can set up an equation using the concept of relative speed.
Let's denote the speed of the boat without a current as "x" km/h.
When the boat travels with the current, its effective speed is the sum of its speed in still water (x km/h) and the speed of the current (3 km/h). Thus, the effective speed is (x + 3) km/h.
Similarly, when the boat travels against the current, its effective speed is the difference between its speed in still water (x km/h) and the speed of the current (3 km/h). Therefore, the effective speed is (x - 3) km/h.
We can relate the distance traveled with the speed and time using the formula: Distance = Speed × Time.
For the trip with the current, the distance traveled is the same as the return trip since it's the same scenic route.
So, if we let "d" represent the distance traveled in km, then for the trip with the current, we have the equation: d = (x + 3) × 3.
Similarly, for the return trip against the current, we have the equation: d = (x - 3) × 7.
Since both equations describe the same distance, we can equate them:
(x + 3) × 3 = (x - 3) × 7.
Now, we can solve this equation to find the value of x.
Expanding the equation, we have:
3x + 9 = 7x - 21.
Moving all the x terms to one side and the constants to the other side, we get:
7x - 3x = 21 + 9.
Simplifying, we have:
4x = 30.
Dividing both sides by 4, we find:
x = 7.5.
Hence, the speed of the boat without a current is 7.5 km/h.
since distance = speed * time,
3(x+3) = 7(x-3)