Hard Calculus Problem? (10 Points for Best Answer)?
I really have no idea where to start with this integration problem (at least that's what I assume it is):
A flea can ride a bike at a rate of 1 mm/sec. The flea starts at point 0 on a ballon being inflated from the moment the flea begins to pedal (at t = 0). Helium is pumped into the balloon at a rate of 1 cubic meters/sec. Will the flea ever make it around?
HINTS:
1. It is unnecessary to determine the rate of progress around the circumference of the expanding balloon.
2. The flea's speed can be thought of as: (a) Its own pedaling, (b) if the flea is a certain distance away from the origin 0, its distance still increases even if it does nothing.
3. If you ignore part (b), just ask yourself: "Is there a time, t, when the guaranteed lower bound of the flea's progress (traveling at a rate of 1 mm/sec) is equal to the length circumference?"
Answers & Comments
Verified answer
Assume the balloon is always a sphere. Then V = (4/3)pi r^3, and C = 2 pi r. That means V = (4/3)pi (C/ 2pi)^3, so V = C^3 / (6pi^2). The derivative gives you dV/dt = [ C^3 / 2pi^2 ] (dC/dt). Note that 1 m^3/sec is equal to 10^9 mm^3/sec, so use that for dV/dt. This gives you an equation in terms of just C and dC/dt, which ultimately shows you that the change in circumference is inversely proportional to the cube of the circumference. You can make use of this in the problem.