Answer:
12.27 ft/s
Explanation:
At 72 ft above the ground, the balloons height increases at a rate of 4ft/s. For 66s, vertical distance moved, y = 4ft/s × 66 s = 264 ft. When the balloon is at 72 ft above the ground, just below it, the bicycle which moves at a rate of 12 ft/s in 66 s, covers a horizontal distance, x = 12ft/s 66 = 792 ft.
The distance between the bicycle and the balloon 66 s later is given by
s = √(x² + (y + 72)²) = √(792² + (264 + 72)²) = √(792² + 336²) = √740160 ft = 860.33 ft
From calculus
The rate of change of the distance between the balloon and bicycle s is obtained by differentiating s with respect to t. So,
ds/dt = (1/s)(xdx/dt + ydy/dt)
dx/dt = 12 ft/s, x = 792 ft, dy/dt = 4 ft/s, y = 264 ft, s = 860.33. These are the values of the variables at t = 66 s.
So, substituting these values into ds/dt, we have
ds/dt = (1/860.33)(792 ft × 12 ft/s + 264 ft × 4ft/s) = (1/860.33)(9504 + 1056) = 10560/860.33 = 12.27 ft/s
