## anonymous 5 years ago 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.

1. anonymous

What is the formula for the volume of a cone?

2. anonymous

V=1/3 pi r^2 d

3. anonymous

V=1/3 pi r^2 d

4. anonymous

correct?

5. anonymous

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.

6. anonymous

...

7. anonymous

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

8. anonymous

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

9. anonymous

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

10. anonymous

And simplify.

11. anonymous

$V = \frac{100\pi}{36d}$

12. anonymous

Now what?

13. anonymous

Now solve for d in terms of V.

14. anonymous

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

15. anonymous

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

16. anonymous

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

17. anonymous

where do I plug my changing rate?

18. anonymous

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

19. anonymous

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

20. anonymous

Don't forget the chain rule!

21. anonymous

How do I take the derivative of that?

22. anonymous

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

23. anonymous

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.

24. anonymous

why did you multiplyby 1/d?

25. anonymous

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

26. anonymous

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

27. anonymous

okay.

28. anonymous

are you back?

29. anonymous

Did you figure it out?