Question

Ms. Davis loves a good cup of coffee. If she puts on a pot of coffee using coneshaped coffee filter of radius 6 cm and depth 10 cm, and the water begins to drip out through the hole at the bottom at a constant rate of 1.5 cm^3 per second, determine how fast the water level is falling when the depth is 8 cm.

Answer 1

What is the formula for the volume of a cone?

Answer 2

V=1/3 pi r^2 d

Answer 3

V=1/3 pi r^2 d

Answer 4

correct?

Answer 5

Probably. I don't remember. So you'll have to come up with another equation of r in terms of d. Probably something from similar triangles.

Answer 6

...

Answer 7

Ok well you know that when d =10, r = 6, And the ratio of the two will be constant throughout the cone.
\[\frac{d}{r} = \frac{10}{6} \implies r = \frac{10}{6d}\]

Answer 8

So now rewrite the volume of the cone formula just in terms of d.

Answer 9

V= 1/3 pi (10/6d)^2*d

Answer 10

And simplify.

Answer 11

\[V = \frac{100\pi}{36d}\]

Answer 12

Now what?

Answer 13

Now solve for d in terms of V.

Answer 14

so i plug in d=10 into my new Volume?

Answer 15

No, because we want to know how fast d is changing at d=8

Answer 16

oh yeah... so I got 25/72 pi

Answer 17

where do I plug my changing rate?

Answer 18

Oh wait. I was wrong. We just need to take the derivative of our new volume equation with respect to time.

Answer 19

\[\frac{d}{dt}V = \frac{d}{dt}[\frac{100\pi}{36d}]\]

Answer 20

Don't forget the chain rule!

Answer 21

How do I take the derivative of that?

Answer 22

\[\frac{d}{dt}V = \frac{100\pi}{36} * \frac{d}{dt}[\frac{1}{d}]\]
\[ = \frac{100\pi}{36} * (ln\ d)*d'\]

Answer 23

And we know that the Volume is changing at a rate of 1.5 per second.

So:
\[1.5 = \frac{100\pi}{36}(ln\ d) * d'\]

Solve for \(d'\) and plug in 8 for d.

Answer 24

why did you multiplyby 1/d?

Answer 25

Because you had \[\frac{100\pi}{36d} = \frac{100\pi}{36} * \frac{1}{d}\]

Answer 26

I've gotta go have dinner. I'll bbl

Answer 27

okay.

Answer 28

are you back?

Answer 29

Did you figure it out?