Calculus Problem (Related Rates)

Question

A cylinder is expanding, and its radius is increasing at a rate of 3 centimeters per minute. The radius of the cylinder is always half of the height of the cylinder; in other words, 2r=h. When the volume of the cylinder is 100 cubic centimeters, how fast is the volume changing?

Answer 1

This is a bit of a long one and I hope I didn't make any silly mistakes along the way. Let's start off with the radius. Since the radius is increasing at 3 cm/min, that means:

dr/dt = 3

Integrating both sides, we have r as a function of t:

r = 3t

I'm assuming the initial radius at time 0 is 0, as this wasn't specified in the problem. Now, the volume of a cylinder is:

V = π r² h

And since we know that 2r = h, we can rewrite this as:

V = 2 π r³

And we can substitute 3t for r to get V as a function of t:

V(t) = 2 π (3t)³ = 54 π t³

We can then use this to find out the time at which the volume is 100:

100 = 54 π t³

t = (50/(27 π))^1/3

Now let's differentiate our volume equation to get an equation for how fast volume is changing with respect to time:

V'(t) = 162 π t²

And finally, we plug in our value of t from above to get the rate of change in volume at that time:

V'((50/(27 π))^1/3) = 162 π (50/(27 π))^2/3 = 18 π (50/π)^2/3

It's a messy answer at the end, so I hope I didn't mess up somewhere along the way. And I don't think it can be simplified any more, but I could be wrong there.