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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.

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  1. anonymous
    • 5 years ago
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    What is the formula for the volume of a cone?

  2. anonymous
    • 5 years ago
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    V=1/3 pi r^2 d

  3. anonymous
    • 5 years ago
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    V=1/3 pi r^2 d

  4. anonymous
    • 5 years ago
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    correct?

  5. anonymous
    • 5 years ago
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    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
    • 5 years ago
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    ...

  7. anonymous
    • 5 years ago
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    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
    • 5 years ago
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    So now rewrite the volume of the cone formula just in terms of d.

  9. anonymous
    • 5 years ago
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    V= 1/3 pi (10/6d)^2*d

  10. anonymous
    • 5 years ago
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    And simplify.

  11. anonymous
    • 5 years ago
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    \[V = \frac{100\pi}{36d}\]

  12. anonymous
    • 5 years ago
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    Now what?

  13. anonymous
    • 5 years ago
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    Now solve for d in terms of V.

  14. anonymous
    • 5 years ago
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    so i plug in d=10 into my new Volume?

  15. anonymous
    • 5 years ago
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    No, because we want to know how fast d is changing at d=8

  16. anonymous
    • 5 years ago
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    oh yeah... so I got 25/72 pi

  17. anonymous
    • 5 years ago
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    where do I plug my changing rate?

  18. anonymous
    • 5 years ago
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    Oh wait. I was wrong. We just need to take the derivative of our new volume equation with respect to time.

  19. anonymous
    • 5 years ago
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    \[\frac{d}{dt}V = \frac{d}{dt}[\frac{100\pi}{36d}]\]

  20. anonymous
    • 5 years ago
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    Don't forget the chain rule!

  21. anonymous
    • 5 years ago
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    How do I take the derivative of that?

  22. anonymous
    • 5 years ago
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    \[\frac{d}{dt}V = \frac{100\pi}{36} * \frac{d}{dt}[\frac{1}{d}]\] \[ = \frac{100\pi}{36} * (ln\ d)*d'\]

  23. anonymous
    • 5 years ago
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    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
    • 5 years ago
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    why did you multiplyby 1/d?

  25. anonymous
    • 5 years ago
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    Because you had \[\frac{100\pi}{36d} = \frac{100\pi}{36} * \frac{1}{d}\]

  26. anonymous
    • 5 years ago
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    I've gotta go have dinner. I'll bbl

  27. anonymous
    • 5 years ago
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    okay.

  28. anonymous
    • 5 years ago
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    are you back?

  29. anonymous
    • 5 years ago
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    Did you figure it out?

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