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## Homework Statement

Here's the problem:

Water is flowing into a water bottle at a rate of 16.5 cm^3/s. The diameter of the bottle varies with its height. How fast is the water level rising when the diameter is 6.30 cm?

## Homework Equations

dV/dt = 16.5

radius = r = 3.15

dh/dt = ?

## The Attempt at a Solution

I know it's a related rates problem, but there's no obvious geometric formula to take the derivative of. Or is there? So I pictured a regular Poland Spring water bottle. I noticed that most sections of the bottle come pretty close to a cylinder. So I guessed that I could approximate the rate with that of a cylinder.

V = pi(h)(r^2)

dV/dt = pi(r^2)(dh/dt)

the radius is the constant throughout a cylinder, so r^2 is a constant.

16.5 = pi(3.15^2)(dh/dt)

dh/dt = 0.529 cm/s

This is the correct answer, but is this the correct way to do this problem. I don't like taking risky guesses and approximations. Luckily it worked this time.