Answer:
Let's solve this problem step by step.
1. Assign variables: Let's denote the speed of the southbound boat as "x" (in mi/h). Since the eastbound boat is traveling 7 mi/h faster, its speed will be "x + 7" (in mi/h).
2. Distance traveled: In 4 hours, the southbound boat will travel a distance of 4x miles, and the eastbound boat will travel a distance of 4(x + 7) miles.
3. Pythagorean theorem: According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In this case, the boats' paths form a right triangle, with the distance between them being the hypotenuse.
4. Applying the Pythagorean theorem: Using the theorem, we can write the equation as follows:
(4x)^2 + (4(x + 7))^2 = 52^2
Simplifying the equation:
16x^2 + 16(x + 7)^2 = 2704
Expanding and simplifying further:
16x^2 + 16(x^2 + 14x + 49) = 2704
16x^2 + 16x^2 + 224x + 784 = 2704
32x^2 + 224x + 784 = 2704
Rearranging the equation to solve for x:
32x^2 + 224x + 784 - 2704 = 0
32x^2 + 224x - 1920 = 0
5. Solving the quadratic equation: We can solve this quadratic equation by factoring or using the quadratic formula. In this case, let's use factoring to simplify the equation:
32(x^2 + 7x - 60) = 0
Factoring further:
32(x + 12)(x - 5) = 0
Setting each factor to zero:
x + 12 = 0 or x - 5 = 0
Solving for x:
x = -12 or x = 5
Since the speed cannot be negative, the speed of the southbound boat is 5 mi/h.
Therefore, the speed of the southbound boat is 5 mi/h.
Step-by-step explanation:
