A kayaker paddled upstream from camp to photograph a waterfall and returns. The kayaker’s speed while traveling upstream is 4 mi/h and the kayaker’s speed while traveling downstream is 7 mi/h. What is the kayaker’s speed in still water? What is the speed of the current?
To find the kayaker's speed in still water and the speed of the current, we can use the concept of relative speed.
Let's assume the kayaker's speed in still water is 'x' mi/h, and the speed of the current is 'c' mi/h.
When the kayaker is paddling upstream (against the current), the effective speed is reduced by the speed of the current. So, the kayaker's speed upstream is (x - c) mi/h.
When the kayaker is paddling downstream (with the current), the effective speed is increased by the speed of the current. So, the kayaker's speed downstream is (x + c) mi/h.
Here's how we can use these equations to solve the problem:
1. When the kayaker paddles upstream, the speed is 4 mi/h, so we have the equation:
(x - c) = 4 ---(eq. 1)
2. When the kayaker paddles downstream, the speed is 7 mi/h, so we have the equation:
(x + c) = 7 ---(eq. 2)
To solve this system of equations, we can use a method called substitution or elimination.
Let's use the substitution method:
From eq. 1, we can express (x - c) in terms of x:
x = 4 + c ---(eq. 3)
Substituting eq. 3 into eq. 2:
(4 + c + c) = 7
4 + 2c = 7
2c = 7 - 4
2c = 3
c = 3/2
c = 1.5
Now, we can substitute the value of c into eq. 3 to find x:
x = 4 + 1.5
x = 5.5
Therefore, the kayaker's speed in still water is 5.5 mi/h, and the speed of the current is 1.5 mi/h.