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A conical salt spreader is spreading salt at a rate of 2 cubic feet per minute. The diameter of the base of the cone is 4 feet and the height of the cone is 5 feet. How fast is the height of the salt in the spreader decreasing when the height of the salt in the spreader (measured from the vertex of the cone upward) is 3 feet? Give your answer in feet per minute. (It will be a positive number since we use the word “decreasing”.)

Mathematics
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always draw the pic...
|dw:1333043238193:dw|the radius of the whole cone is 2 and the height is 5

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Other answers:

|dw:1333043327825:dw|at some given moment the cone of remaining salt is radius r and height h from this we can find similar triangles....
|dw:1333043429599:dw|so we get the proportion\[\frac25=\frac rh\implies r=\frac25h\]now we clearly need the formula for the volume of the cone, which is\[V=\frac13\pi r^2h\]to get this all in terms of one variable we will note the fact we got from the similar triangles....
\[V(h)=\frac13\pi(\frac25h)^2h=\frac4{75}\pi h^3\]can you differentiate this with respect to time?
.44
I'm trying to figure out what you did here can you just show me what you get after differentiating wrt time?
2*(75)/(4)*(3.14)*(27)=.44
don't plug in any numbers yet
I'm using my notes. .44 is the correct answer.
ok, I'm not using a calculator, so I didn't know that I just wanted to make sure you knew how to differentiate this with respect to time
I don't suppose you feel like humoring me and showing me what you get after differentiating \(before\) plugging in the numbers?
yeah... h^3 becomes 3h^2 then 3(3)^2 =27
no that's not how you differentiate wrt \(time\) you differentiated wrt to height, which makes this problem unsolvable
I mean, where did you get 0.44 aside from the fact that you already knew the answer? you forgot to differentiate \(implicitly\) h, which must be done because it is a function of time (we want rate change of height in units of time, dh/dt)
i didnt know the answer
so then how did you get 0.44
2*(75)/(4)*(3.14)*(27)=.44 i got it by doing this ^^^^
ok, see I couldn't decipher that you did it right, because you plugged in the numbers already, so my brain can't read the problem backwards that fast!
the way I was writing... so we have\[V=\frac4{75}\pi h^3\]differentiating with respect to \(time\) we get\[{dV\over dt}=\frac4{25}\pi h^2{dh\over dt}\]so we solve for dh/dt, which you did but when you plugged in the numbers it was hard for me to see that you had done that bit correctly
well, nice job then :D so for future reference: these cone problems are always done using the similar triangle trick that is sort of the motto here see ya 'round
alrighty thanks :)

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