At noon, you and I start at the same place, and then you head east at a speed of 5 kilometers per hour, and I head north at a speed of 4 kilometers per hour. After two hours, how fast are we moving apart, in kilometers per hour?

To find out how fast we are moving apart, we can use the Pythagorean theorem. The Pythagorean theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.

In this case, our speeds represent the lengths of the sides of a right triangle. So, using the Pythagorean theorem, we can calculate the speed at which we are moving apart.

Let's call the distance you have traveled after 2 hours as "distance_east", and the distance I have traveled after 2 hours as "distance_north".

Since you are moving east at 5 kilometers per hour for 2 hours, your distance_east can be calculated as:
distance_east = speed_east * time = 5 km/h * 2 h = 10 km.

Similarly, since I am moving north at 4 kilometers per hour for 2 hours, my distance_north can be calculated as:
distance_north = speed_north * time = 4 km/h * 2 h = 8 km.

Now, we can use the Pythagorean theorem to find the total distance between our positions after 2 hours:
distance_apart = sqrt(distance_east^2 + distance_north^2)
distance_apart = sqrt(10 km^2 + 8 km^2)
distance_apart = sqrt(100 km^2 + 64 km^2)
distance_apart = sqrt(164 km^2)
distance_apart ≈ 12.81 km (rounded to two decimal places)

Therefore, we are moving apart at a speed of approximately 12.81 kilometers per hour after two hours.

To determine how fast we are moving apart, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

In this scenario, the two sides of the right triangle represent the distances you and I have traveled after two hours. Let's denote your distance as x and my distance as y.

Since you have been heading east at a speed of 5 kilometers per hour for two hours, your distance traveled can be calculated as:

Distance you traveled (x) = speed * time = 5 km/h * 2 h = 10 km.

Similarly, since I have been heading north at a speed of 4 kilometers per hour for two hours, my distance traveled can be calculated as:

Distance I traveled (y) = speed * time = 4 km/h * 2 h = 8 km.

Now, we can use the Pythagorean theorem to find the distance between our current positions, which represents how fast we are moving apart. The equation is:

Distance^2 = x^2 + y^2.

Substituting the values we found earlier, we have:

Distance^2 = 10^2 + 8^2 = 100 + 64 = 164.

To find the actual distance, we take the square root of both sides:

Distance = √164 ≈ 12.806 km.

Therefore, after two hours, we are moving apart at a speed of approximately 12.806 kilometers per hour.