Question

Water is poured into a conical paper cup at the rate of 3/2 in3/sec. If the cup is 6 inches tall and the top
has a radius of 3 inches, how fast is the water level rising when the
water is 2 inches deep?

The water level is rising at a rate of what?

Answer 1

lordamercy any idea what the equation for the volume of a cone is?

Answer 2

(pi*r^2*h)/3

Answer 3

\[\frac{ 1 }{ 3 }*\pi*r ^{2}*h\]

Answer 4

yeah so we need some sort of expression for the volume in terms of \(h\) i guess we have to use similar triangles or something like that

Answer 5

yea I got that r=h/3

Answer 6

ok maybe i made a mistake, because it is late, but i get \(\frac{r}{h}=\frac{3}{6}=\frac{1}{2}\) making \(h=2r\)

Answer 7

i could be wrong, please let me know if i made a mistake
if not we can finish fairly easily

Answer 8

lets assume for the sake of argument that i am right
then you have
\[V=\frac{\pi r^2h}{3}\] putting \(r=\frac{h}{2}\) gives
\[V(h)=\frac{\pi h^3}{12}\]

Answer 9

taking derivatives gives
\[V'=\frac{\pi h^2}{4}h'\]

Answer 10

you are told \(V'=\frac{2}{3}\) and that \(h=2\) solve for \(h'\)

Answer 11

3/2

Answer 12

oh right \(V'=\frac{3}{2}\) right

Answer 13

I think it's h=3r

Answer 14

but we have to solve in terms of h, I think

Answer 15

yeaaaa it's r=h/3

Answer 16

not sure where the 3 comes from
the radius is 3, the height is 6
the radius is r, the height is h
that means i think that \(h=2r\) oh and i messed up

Answer 17

no no i didn't we want \(h\) not \(r\) so \(r=\frac{h}{2}\)

Answer 18

you may be right, but can you explain why \(r=\frac{h}{3}\) ?

Answer 19

oh goooof I can't lol sorry i divided 3 by 6 and thought it's 1/3, so sorry

Answer 20

so the final answer is 3/2*pi???

Answer 21

i didn't get to the final answer yet, but be can do it now

Answer 22

\[V'=\frac{\pi h^2}{4}h'\] or
\[h'=\frac{4V'}{\pi h^2}\]

Answer 23

put \(h=2, V'=\frac{3}{2}\) and you are done

Answer 24

no that is wrong

Answer 25

now i think it is
\[h'=\frac{3}{2\pi}\]

Answer 26

Thank YOU!