Final answer:
The train and the hot air balloon are moving away from each other at approximately 50.99 mph, calculated using the Pythagorean theorem to combine their perpendicular velocities.
Step-by-step explanation:
If a train is moving at a constant rate of 50 mph and a hot air balloon is rising at a constant rate of 10 mph, and both start moving at the same time from the same point, we can determine the rate at which they are moving away from one another by considering their movement in two perpendicular dimensions. The train moves horizontally while the hot air balloon rises vertically. To find the rate at which the distance between the two increases, we use the Pythagorean theorem because their paths form a right triangle.
The rate at which the train moves away from the starting point is the horizontal leg of the triangle (50 mph), and the rate at which the hot air balloon rises is the vertical leg of the triangle (10 mph). The hypotenuse of this right triangle represents the rate at which the two are separating from each other. Using the Pythagorean theorem:
Separation rate = √(horizontal rate^2 + vertical rate^2)
= √(50^2 + 10^2)
= √(2500 + 100)
= √(2600)
= ≈50.99 mph
Therefore, the train and the hot air balloon are moving away from one another at approximately 50.99 mph.