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anonymous

  • one year ago

Water Runs into a conical tank at the rate of 3 feet^3/min. the tank stands pointing down and has a height of 15ft and a base radius of 4ft. How fast is the water level rising when the water is 8 feet deep.

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  1. anonymous
    • one year ago
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    |dw:1443496421968:dw|

  2. anonymous
    • one year ago
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    not sure where to go from there

  3. Vocaloid
    • one year ago
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    |dw:1443496845151:dw| first step is to calculate r' (the value I marked), do you know how to do that?

  4. anonymous
    • one year ago
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    yea i think i do its, is this right ? \[\frac{ r }{ h } = \frac{ 4 }{ 15 } \] \[r = \frac{ 4h }{ 15 }\]

  5. Vocaloid
    • one year ago
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    almost, we want the radius when h = 8, so you can plug in h = 8 into your proportion

  6. anonymous
    • one year ago
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    32/15 = 2.1333

  7. Vocaloid
    • one year ago
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    uh, lets leave it as a fraction for now (32/15) now, we just figured out that r = 4h/15, we can substitute that into the volume equation

  8. Vocaloid
    • one year ago
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    V = (1/3)(pi)(r^2)(h) = (1/3)(pi)(4h/15)^2*(h) now differentiate that with respect to h

  9. Vocaloid
    • one year ago
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    *with respect to t

  10. anonymous
    • one year ago
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    okay so \[V = \frac{ 1 }{ 3 } \pi (\frac{ 32 }{ 15 }) ^2 h\]

  11. Vocaloid
    • one year ago
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    actually, forget 32/15, we don't really need that (my mistake)

  12. Vocaloid
    • one year ago
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    |dw:1443497509872:dw|

  13. anonymous
    • one year ago
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    wait so do we treat the r in the function as a constant = to 32/15 ?

  14. anonymous
    • one year ago
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    ah i see okay

  15. Vocaloid
    • one year ago
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    right, we re-wrote r in terms of h, so we've already accounted for r (since r changes proportionally to h)

  16. anonymous
    • one year ago
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    so dv/dt = 2pi/3 (4h/15) h dh/dt ?

  17. anonymous
    • one year ago
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    is that right?

  18. Vocaloid
    • one year ago
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    not quite

  19. Vocaloid
    • one year ago
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    |dw:1443497761147:dw|

  20. Vocaloid
    • one year ago
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    then plug in h = 8 and dv/dt = 3

  21. anonymous
    • one year ago
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    wait why is h cubed, and also i thought it was 4h/15 not 4/15

  22. Vocaloid
    • one year ago
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    |dw:1443498112065:dw|

  23. Vocaloid
    • one year ago
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    remember your exponent rules

  24. anonymous
    • one year ago
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    :P thats useful

  25. anonymous
    • one year ago
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    okay and where did the 1/3 go?

  26. anonymous
    • one year ago
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    okay nvm i see

  27. anonymous
    • one year ago
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    the power rule for the h^3 canceled it out

  28. Vocaloid
    • one year ago
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    remember your derivative rules|dw:1443498227202:dw|

  29. Vocaloid
    • one year ago
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    yeah that

  30. anonymous
    • one year ago
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    yep and the answer is .2098

  31. anonymous
    • one year ago
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    okay thank you very much!!!

  32. Vocaloid
    • one year ago
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    would be best to write it as a fraction in terms of pi, but yeah

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