anonymous
  • anonymous
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.
Mathematics
katieb
  • katieb
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anonymous
  • anonymous
|dw:1443496421968:dw|
anonymous
  • anonymous
not sure where to go from there
Vocaloid
  • Vocaloid
|dw:1443496845151:dw| first step is to calculate r' (the value I marked), do you know how to do that?

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anonymous
  • anonymous
yea i think i do its, is this right ? \[\frac{ r }{ h } = \frac{ 4 }{ 15 } \] \[r = \frac{ 4h }{ 15 }\]
Vocaloid
  • Vocaloid
almost, we want the radius when h = 8, so you can plug in h = 8 into your proportion
anonymous
  • anonymous
32/15 = 2.1333
Vocaloid
  • Vocaloid
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
Vocaloid
  • Vocaloid
V = (1/3)(pi)(r^2)(h) = (1/3)(pi)(4h/15)^2*(h) now differentiate that with respect to h
Vocaloid
  • Vocaloid
*with respect to t
anonymous
  • anonymous
okay so \[V = \frac{ 1 }{ 3 } \pi (\frac{ 32 }{ 15 }) ^2 h\]
Vocaloid
  • Vocaloid
actually, forget 32/15, we don't really need that (my mistake)
Vocaloid
  • Vocaloid
|dw:1443497509872:dw|
anonymous
  • anonymous
wait so do we treat the r in the function as a constant = to 32/15 ?
anonymous
  • anonymous
ah i see okay
Vocaloid
  • Vocaloid
right, we re-wrote r in terms of h, so we've already accounted for r (since r changes proportionally to h)
anonymous
  • anonymous
so dv/dt = 2pi/3 (4h/15) h dh/dt ?
anonymous
  • anonymous
is that right?
Vocaloid
  • Vocaloid
not quite
Vocaloid
  • Vocaloid
|dw:1443497761147:dw|
Vocaloid
  • Vocaloid
then plug in h = 8 and dv/dt = 3
anonymous
  • anonymous
wait why is h cubed, and also i thought it was 4h/15 not 4/15
Vocaloid
  • Vocaloid
|dw:1443498112065:dw|
Vocaloid
  • Vocaloid
remember your exponent rules
anonymous
  • anonymous
:P thats useful
anonymous
  • anonymous
okay and where did the 1/3 go?
anonymous
  • anonymous
okay nvm i see
anonymous
  • anonymous
the power rule for the h^3 canceled it out
Vocaloid
  • Vocaloid
remember your derivative rules|dw:1443498227202:dw|
Vocaloid
  • Vocaloid
yeah that
anonymous
  • anonymous
yep and the answer is .2098
anonymous
  • anonymous
okay thank you very much!!!
Vocaloid
  • Vocaloid
would be best to write it as a fraction in terms of pi, but yeah

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