Final answer:
The rate of change of the distance between the motorcycle and the balloon 10 seconds later involves the use of the Pythagorean theorem and differentiation to find the rate at which the distance is changing at that specific moment in time.
Step-by-step explanation:
The problem is based on rate of change and can be solved using the Pythagorean theorem and differentiation. Let's define our variables where x(t) is the horizontal distance of the motorcycle from the point on the ground below the balloon, y(t) is the height of the balloon above the ground, and d(t) is the distance between the motorcycle and the balloon. Ten seconds later, the motorcycle's horizontal distance from the starting point is x(10) = 660 ft (since 66 ft/s * 10 s = 660 ft) and the height of the balloon is y(10) = 260 ft (since it started at 160 ft and rose at 10 ft/s for 10 s).
We then apply the Pythagorean theorem to find d(t):
d(t)² = x(t)² + y(t)². Differentiate both sides with respect to time t to find the rate of change of d(t) at t = 10 seconds. Using the given rates of change for x(t) and y(t), we can solve for dd/dt when t = 10.