Two fishing boats depart a harbor at the same time, one traveling east, the other south. The eastbound boat travels at a speed 1 mi/h faster than the southbound boat. After 5 h the boats are 25 mi apart. Find the speed of the southbound boat.
Let $x$ be the speed of the southbound boat in miles per hour. Then the eastbound boat has speed $x + 1$ miles per hour. After 5 hours, the southbound boat travels $5x$ miles and the eastbound boat travels $5(x + 1)$ miles. Therefore, by the Pythagorean Theorem,
\[25^2 = (5x)^2 + (5(x + 1))^2.\]Simplifying the expression on the right, we find
\[625 = 25x^2 + 25(x^2 + 2x + 1).\]This simplifies to $50x^2 + 50x - 600 = 0.$ Dividing by 50, we get $x^2 + x - 12 = 0,$ which factors as $(x - 3)(x + 4) = 0.$ Hence, $x = \boxed{3}.$
Let's call the speed of the southbound boat "x" mi/h.
Since the eastbound boat travels at a speed 1 mi/h faster, its speed would be "x + 1" mi/h.
After 5 hours, the southbound boat would have traveled a distance of 5x miles.
The eastbound boat would have traveled a distance of 5(x + 1) miles.
Since the boats are 25 miles apart, we can create a right-angled triangle where the distance traveled by the southbound boat is one side, the distance traveled by the eastbound boat is another side, and the distance between the boats is the hypotenuse.
According to the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides, the equation is:
(5x)^2 + 5(x + 1)^2 = 25^2
Simplifying the equation:
25x^2 + 5(x^2 + 2x + 1) = 625
25x^2 + 5x^2 + 10x + 5 = 625
30x^2 + 10x + 5 = 625
30x^2 + 10x - 620 = 0
Dividing the equation by 10:
3x^2 + x - 62 = 0
We can solve this quadratic equation using factoring, completing the square, or the quadratic formula.
Using the quadratic formula, where ax^2 + bx + c = 0:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 3, b = 1, and c = -62:
x = (-1 ± √(1^2 - 4(3)(-62)) / (2(3))
x = (-1 ± √(1 + 744)) / 6
x = (-1 ± √745) / 6
Since speed cannot be negative, we discard the negative solution:
x = (√745 - 1) / 6
Therefore, the speed of the southbound boat is approximately (√745 - 1) / 6 mi/h.