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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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.

Mathematics
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What is the formula for the volume of a cone?
V=1/3 pi r^2 d
V=1/3 pi r^2 d

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correct?
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.
...
Ok well you know that when d =10, r = 6, And the ratio of the two will be constant throughout the cone. \[\frac{d}{r} = \frac{10}{6} \implies r = \frac{10}{6d}\]
So now rewrite the volume of the cone formula just in terms of d.
V= 1/3 pi (10/6d)^2*d
And simplify.
\[V = \frac{100\pi}{36d}\]
Now what?
Now solve for d in terms of V.
so i plug in d=10 into my new Volume?
No, because we want to know how fast d is changing at d=8
oh yeah... so I got 25/72 pi
where do I plug my changing rate?
Oh wait. I was wrong. We just need to take the derivative of our new volume equation with respect to time.
\[\frac{d}{dt}V = \frac{d}{dt}[\frac{100\pi}{36d}]\]
Don't forget the chain rule!
How do I take the derivative of that?
\[\frac{d}{dt}V = \frac{100\pi}{36} * \frac{d}{dt}[\frac{1}{d}]\] \[ = \frac{100\pi}{36} * (ln\ d)*d'\]
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.
why did you multiplyby 1/d?
Because you had \[\frac{100\pi}{36d} = \frac{100\pi}{36} * \frac{1}{d}\]
I've gotta go have dinner. I'll bbl
okay.
are you back?
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