Final answer:
The rate of change of the distance between the motorcycle and the balloon after 7 seconds is approximately 45.67 ft/s, which doesn't match any of the given options. This was calculated using the Pythagorean theorem for both the distance and the rate.
Step-by-step explanation:
To find the rate of change of the distance between the motorcycle and the balloon after 7 seconds, we'll need to use the Pythagorean theorem to calculate the distance between them. Initially, the balloon is 180 ft above the ground and the motorcycle passes beneath it at 44 ft/s. After 7 seconds, the balloon has risen 12 ft/s * 7 s = 84 ft, so it is now 180 ft + 84 ft = 264 ft above the ground. In the same time, the motorcycle has traveled 44 ft/s * 7 s = 308 ft horizontally.
Using the Pythagorean theorem (a2 + b2 = c2), we find the diagonal distance (c) as:
c2 = 2642 + 3082
c = √(2642 + 3082)
c ≈ 405.66 ft (rounded to two decimal places)
The rate of change of this distance can be approximated by the relative speeds of the bike and balloon. Since they're moving perpendicularly to each other, we'll use the Pythagorean theorem again:
Rate = √(442 + 122)
Rate ≈ 45.67 ft/s (rounded to two decimal places)
Therefore, after 7 seconds the rate of change of the distance between the motorcycle and the balloon is approximately 45.67 ft/s, which doesn't match any of the options given. It seems there might be a mistake in the provided options.