1) Variables and Table:
Let's label the speed of the kayaker as "k" and the speed of the current as "c". We can use the following table to organize the information:
| Distance | Rate | Time |
|------------|--------|--------|
| 1 mile | k - c | t1 |
| 1 mile | k + c | t2 |
| 2 miles | | 3h20m |
Note that we use "t1" and "t2" to represent the time it takes to travel one mile in each direction, since the kayaker is traveling at a different rate relative to the current in each direction.
2) Quadratic Equation:
To solve for "k", we can use the formula:
distance = rate x time
For the first leg of the trip, we have:
1 = (k - c) x t1
Solving for t1, we get:
t1 = 1 / (k - c)
For the second leg of the trip, we have:
1 = (k + c) x t2
Solving for t2, we get:
t2 = 1 / (k + c)
Since the total time of the trip is 3 hours 20 minutes, or 3.33 hours, we can write:
t1 + t2 = 3.33
Substituting the expressions for t1 and t2, we get:
1/(k-c) + 1/(k+c) = 3.33
Multiplying both sides by (k-c)(k+c), we get:
(k+c) + (k-c) = 3.33(k-c)(k+c)
Simplifying, we get:
2k = 3.33(k^2 - c^2)
Multiplying out the right side, we get:
2k = 3.33k^2 - 3.33c^2
Rearranging and setting the equation equal to zero, we get:
3.33k^2 - 2k - 3.33c^2 = 0
This is a quadratic equation in "k".
3) Solving the Equation:
We can solve this quadratic equation using the quadratic formula:
k = (-b ± sqrt(b^2 - 4ac)) / 2a
where a = 3.33, b = -2, and c = -3.33c^2. Substituting these values, we get:
k = (-(-2) ± sqrt((-2)^2 - 4(3.33)(-3.33c^2))) / 2(3.33)
Simplifying, we get:
k = (2 ± sqrt(4 + 44.286c^2)) / 6.66
4) Explanation of Results:
The quadratic equation has two solutions for "k", but one of them is negative and therefore not physically meaningful. The other solution gives the speed of the kayaker relative to the water:
k = (2 + sqrt(4 + 44.286c^2)) / 6.66
This equation shows that the speed of the kayaker depends on the speed of the river current. As the current gets stronger (i.e., as "c" increases), the kayaker needs to paddle faster to maintain a constant speed relative to the water. Conversely, if the current is weaker, the kayaker can paddle more slowly and still maintain the same speed.