A motorboat can maintain a constant speed of 26 miles per hour relative to the water. The boat makes a trip upstream to a certain point in 35 minutes; the return trip takes 17 minutes. What is the speed of the river current?
To solve this problem, we can use the concept of relative speed. Let's assume that the speed of the river current is 'x' miles per hour.
During the upstream trip, the boat is moving against the current, so the effective speed will be the difference between the boat's speed and the river's speed: 26 - x.
During the downstream trip, the boat is moving with the current, so the effective speed will be the sum of the boat's speed and the river's speed: 26 + x.
Now, we can use the formula Distance = Speed × Time to find the distances traveled in each direction:
1. Upstream trip:
Distance = (26 - x) × (35/60) (converting 35 minutes to hours)
Distance = (26 - x) × (7/12) (simplifying the fraction)
2. Downstream trip:
Distance = (26 + x) × (17/60) (converting 17 minutes to hours)
Distance = (26 + x) × (17/60) (simplifying the fraction)
Since the distances covered in both directions are the same, we can equate the two distances:
(26 - x) × (7/12) = (26 + x) × (17/60)
To solve this equation, we can cross-multiply and then simplify:
7(26 - x) = 12(26 + x)
182 - 7x = 312 + 12x
Combining like terms:
-7x - 12x = 312 - 182
-19x = 130
To find the value of x, divide both sides of the equation by -19:
x = -130/19
So, the speed of the river current is approximately -6.842 miles per hour (rounded to three decimal places).
Note: The negative sign indicates that the current is flowing in the opposite direction of the boat.
To find the speed of the river current, we can use the concept of relative velocity and the formula:
Speed of the boat = Speed of the water current + Speed of the boat relative to the water
Let's assume the speed of the river current is "x" miles per hour.
For the upstream trip, the boat is going against the current, which means the effective speed of the boat will be reduced. We can calculate it using:
Speed of the boat upstream = Speed of the water current - Speed of the boat relative to the water
Given that the boat takes 35 minutes to reach the certain point, we need to convert it to hours:
35 minutes = 35/60 = 7/12 hours
Similarly, for the downstream trip, the boat is going with the current, so the effective speed of the boat will be increased. We can calculate it using:
Speed of the boat downstream = Speed of the water current + Speed of the boat relative to the water
Given that the return trip takes 17 minutes, we convert it to hours:
17 minutes = 17/60 = 17/60 hours
Now, let's formulate the equations based on the given information:
Upstream trip: 26 = x - Speed of the boat relative to the water (Equation 1)
Downstream trip: 26 = x + Speed of the boat relative to the water (Equation 2)
We can solve these equations simultaneously to find the value of x, which represents the speed of the river current.
To eliminate the variable "Speed of the boat relative to the water," we can add Equation 1 and Equation 2:
26 + 26 = (x - Speed of the boat relative to the water) + (x + Speed of the boat relative to the water)
Simplifying this expression, we get:
52 = 2x
Now, divide both sides by 2:
52/2 = x
x = 26
Therefore, the speed of the river current is 26 miles per hour.
To summarize:
To find the speed of the river current, we used the concept of relative velocity. We formulated equations based on the given information, solved the equations, and found that the speed of the river current is 26 miles per hour.