A hot-air balloon is 150 feet above the ground when a motorcycle passes directly beneath it (traveling in a straight line on a horizontal road). The motorcycle is traveling at 40 mph. If the balloon is rising vertically at a rate of 10 ft/sec, what is the rate of change of the distance between the balloon and the motorcycle ten seconds later?
To find the rate of change of the distance between the balloon and the motorcycle ten seconds later, we need to determine how their distances change over time separately.
Let's first analyze the rate of change of the motorcycle's distance from the ground. The motorcycle is traveling at a constant speed of 40 mph, which is equivalent to 40 * (5280 / 3600) = 58.67 ft/sec. Since the motorcycle is traveling in a straight line, its distance from the ground does not change.
Now, let's focus on the rate of change of the balloon's distance from the ground. The balloon is rising vertically at a rate of 10 ft/sec. So, after 10 seconds, the balloon will be 10 * 10 = 100 feet higher than initially. Therefore, its distance from the ground will be 150 + 100 = 250 feet.
To find the rate of change of the distance between the balloon and the motorcycle, we need to subtract their distances from the ground. Thus, the rate of change of the distance between them will be 250 ft - 0 ft = 250 ft.
Therefore, the rate of change of the distance between the balloon and the motorcycle ten seconds later is 250 ft.