Cone - V=1/3 pi r^2 h however I can't differentiate

[markosheehan]

A vessel in the shape of a cone is standing on its apex. Water flows in at a steady rate, of 1m^3 per minute. The vessel has a height of 2m and a diameter of 2m when the vessel is 1/8 full find the rate at which water is rising

V=1/3 pi r^2 h however i can't differentiate this

[markosheehan]

i actually worked it out using dv/dt=dv/dh*dh/dt the answer i got was pi/4.im stuck on the next part though. what is the rate at which the free surface area of the water is increasing when the cone is 1/8th full

[Guest]

It really annoys me that we cannot "edit" a post on this forum!

Yes, the rate at which the water is rising is dh/dt. However, "[tex]V= \frac{1}{3}\pi r^2h[/tex]" is not enough. You also need to know how "h" varies with "r". [tex]\frac{dV}{dh}= \frac{1}{3}\pi\left(2rh\frac{dr}{dt}+ r^2\frac{dh}{dt}\right)[/tex]. In this problem you are told that "The vessel has a height of 2m and a diameter of 2m" which means that whatever the height of water in the vessel is, the radius is equal to it: r= h. We [tex]\frac{dr}{dt}= \frac{dh}{dt}[/tex]. Making those changes in the derivative, [tex]\frac{dV}{dt}= \frac{1}{3}\pi\left(2h^2\frac{dh}{dt}+ r^2\frac{dh}{dt}= \frac{1}{3}\pi )(3h^2)\frac{dh}{dt}\right)= \pi h^2 \frac{dh}{dt}[/tex]. We could also have done this by setting r= h initially: [tex]V= \frac{1}{3}\pi h^3[/tex] so [tex]\frac{dV}{dt}= \pi h^2\frac{dh}{dt}[/tex].

We are told that water is coming in at 1 cubic meter per second so [tex]\frac{dV}{dt}= 1[/tex]. We are asked how the level of the water is increasing, [tex]\frac{dh}{dt}[/tex], when the cone is 1/8 full. The full volume is [tex]\frac{1}{3}\pi (2)^3= frac{8}{3}\pi[/tex] and 1/8 of that is [tex]\frac{1}{3}\pi[/tex] cubic meters. Then [tex]\frac{1}{3}\pi h^3= \frac{1}{3}\pi[/tex]. h= r= 1 and we have [tex]\frac{dV}{dt}= 1= \pi (1)^2\frac{dh}{dt}[/tex] so [tex]\frac{dh}{dt}= \frac{1}{\pi}[/tex] meters per second, not [tex]\frac{\pi}{4}[/tex]. How did you get that?

For the second part, since r= h at every height, the surface of the water in the cone is given by [tex]A= \pi r^2= \pi h^2[/tex] so [tex]\frac{dA}{dt}= 2\pi h \frac{dh}{dt}[/tex].

[Math Tutor]

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[Guest]

Thanks. I am "registered" but I seldom bother to "log in"! I will have to start.

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