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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.
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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anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0What is the formula for the volume of a cone?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Probably. 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.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Ok well you know that when d =10, r = 6, And the ratio of the two will be constant throughout the cone. \[\frac{d}{r} = \frac{10}{6} \implies r = \frac{10}{6d}\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0So now rewrite the volume of the cone formula just in terms of d.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0V= 1/3 pi (10/6d)^2*d

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[V = \frac{100\pi}{36d}\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Now solve for d in terms of V.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0so i plug in d=10 into my new Volume?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0No, because we want to know how fast d is changing at d=8

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0oh yeah... so I got 25/72 pi

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0where do I plug my changing rate?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Oh wait. I was wrong. We just need to take the derivative of our new volume equation with respect to time.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[\frac{d}{dt}V = \frac{d}{dt}[\frac{100\pi}{36d}]\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Don't forget the chain rule!

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0How do I take the derivative of that?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[\frac{d}{dt}V = \frac{100\pi}{36} * \frac{d}{dt}[\frac{1}{d}]\] \[ = \frac{100\pi}{36} * (ln\ d)*d'\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0And 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.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0why did you multiplyby 1/d?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Because you had \[\frac{100\pi}{36d} = \frac{100\pi}{36} * \frac{1}{d}\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I've gotta go have dinner. I'll bbl

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Did you figure it out?
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