Two boats leave the same marina. One heads north, and the other heads east. After some time, the northbound boat has traveled 16 miles. If the two boats are 20 miles apart, how far has the eastbound boat traveled?

This problem can be solved using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Here, the distance between the two boats forms the hypotenuse, the distance the northbound boat has traveled forms one side, and the distance the eastbound boat has traveled forms the other side.

Given:

• The northbound boat has traveled 16 miles (one side of the triangle)
• The two boats are 20 miles apart (the hypotenuse of the triangle)

We need to find:

• The distance the eastbound boat has traveled (the other side of the triangle)

According to the Pythagorean theorem:

(side1)^2 + (side2)^2 = (hypotenuse)^2

Let's plug in the given values:

(16 miles)^2 + (eastbound distance)^2 = (20 miles)^2

Solving this equation will give us the distance the eastbound boat has traveled.

256 (miles)^2 + (eastbound distance)^2 = 400 (miles)^2

Subtract 256 (miles)^2 from both sides to isolate (eastbound distance)^2 on one side of the equation:

(eastbound distance)^2 = 400 (miles)^2 - 256 (miles)^2 (eastbound distance)^2 = 144 (miles)^2

Finally, take the square root of both sides to solve for the eastbound distance:

eastbound distance = sqrt(144 (miles)^2) eastbound distance = 12 miles

So, the eastbound boat has traveled 12 miles.