Is there an alternative method to solve this problem?

[asap9993]

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.

 

[Delphi51]

I don't care for the V = pi(h)(r^2).
Better to picture an infinitesimally thick layer of water of height dh, which has volume dV = πr²*dh. That gives you an expression for dh/dV that you can put together with the given value for dV/dt to get dh/dt. Same answer.

 

[tiny-tim]

hi asap9993!

(have a pi: π and try using the X² tag just above the Reply box )

here's a reasonably systematic way …

the basic formula for volume is V = ∫_x=0^h A dx (A = area, h = height).

So dV/dh = … ? ​

 

[asap9993]

To tiny-tim,

I don't quite understand why we are concentrating on the rate of change of the volume with respect to height instead of time. If you define volume as that integral then,

dV/dh = A = pi(r^2) = pi(3.15^2) = 31.17 cm^2

I guess we could find how the height is changing with respect to time by the chain rule:

dh/dt = (dV/dt)(dh/dV)

dh/dt = (16.5)(1/31.17) = 0.529 cm/s

Is that it?

 

[tiny-tim]

hi asap9993!

I don't quite understand why we are concentrating on the rate of change of the volume with respect to height instead of time.
…
I guess we could find how the height is changing with respect to time by the chain rule:
…
Is that it?

yes, that's right

do the obvious first (volume vs height), then use the chain rule to adapt it to the question asked