A kayaker paddles upstream for 1.5 hours then turns his kayak around and returns to his tent in 1 hour. He travels 3 miles each way. What is the rate of the rivers current?
speed up = u -c
speed down = u+c
d = speed * time
3 = (u-c)1.5
3 = (u+c)1
1.5 u - 1.5 c = 3
1.5 u + 1.5 c = 4.5
--------------------- add
3 u = 7.5
u = 7.5/3
3 = 7.5/3 + c
c = (9-7.5)/3 = 2.5/3 = .83333333333333333333333333
To find the rate of the river's current, we need to understand the relationship between time, distance, and speed. Let's break down the problem into two parts: paddling upstream and paddling downstream.
First, let's consider paddling upstream. The kayaker spends 1.5 hours and covers a distance of 3 miles. Since the current is against him, his effective speed is reduced by the speed of the river's current. Let's call this speed "V" (which is what we want to find).
We can use the formula: Distance = Speed × Time, rearranged to solve for speed: Speed = Distance / Time.
For the upstream trip, the kayaker's effective speed will be V (kayaker's speed relative to the ground) minus the speed of the river's current. So, the kayaker's effective speed upstream can be written as V - C (where C is the rate of the river's current).
We can now plug in the values and write the equation for paddling upstream:
3 miles = (V - C) × 1.5 hours
Now let's consider the downstream trip. The kayaker takes only 1 hour to travel the same 3 miles, but this time, the river's current is in his favor. The kayaker's effective speed downstream will be V (kayaker's speed relative to the ground) plus the speed of the river's current. So, the kayaker's effective speed downstream can be written as V + C.
Using the formula again, the equation for paddling downstream can be written as:
3 miles = (V + C) × 1 hour
Now we have a system of two equations:
(1) 3 = (V - C) × 1.5
(2) 3 = (V + C) × 1
We can solve this system of equations to find the value of V (kayaker's speed relative to the ground) and C (rate of the river's current).
To simplify things, let's start by dividing equation (1) by 1.5:
2 = V - C
Now let's solve for V in terms of C by subtracting C from both sides:
V = C + 2
Next, we can substitute V in equation (2) with C + 2:
3 = (C + 2 + C) × 1
Simplifying equation (2):
3 = (2C + 2) × 1
3 = 2C + 2
Now, subtracting 2 from both sides of equation (2):
1 = 2C
To isolate C (rate of the river's current), divide both sides of equation (2) by 2:
C = 1/2
Therefore, the rate of the river's current is 0.5 miles per hour.