application problem in derivatives

[ksguru]

hi

i need help to solve this problem.

An empty hollow cone of radius 'a' and height '4a' is held vertex down and water is poured in a rate of 8pi cm3/s. Find the rate at which the depth of water is increasing after 25 secs.

dV/dt = 8 pi(given)

RTF dh/dt =? where 'h' is the height i took after 25 secs.

V= volumeof the cone- vol of the cone at 25secs (that is 2 sym cones)

Did I start off right?

KIndly help me understand this problem

thanks.

kal

[aldld]

I'm not quite sure if this is right or not, but I'll give it a shot...

So you're given dV/dt = 8pi. Solving for V you get:

Recall the formula for the volume of a cone with radius r and height h:

So now you have two expressions for V, so you can substitute them into each other to get an equation involving r, and t.

Let's say we want to write r as a function of h. Because the radius of the cone is a and the height is 4a, we get r = h/4. Substitute this into the equation for the previous step and we get an equation involving h and t:

You can solve this equation for h to get height as a function of time, and then take the derivative of that function (or skip solving for h altogether and use implicit differentiation) and substitute in 25s for your time to get your answer.

Sorry it was a bit rushed, if you want any clarification or if you want me to go into more detail in any steps I'd be happy to do so. (Also it's possible that I may have made a mistake somewhere, so if you see anything wrong with this (or if you have a better method) please point it out )

[ksguru]

HI

I did proceed in a similar way. But I reckon there is an error in equating the 'V'.

It must be * 1/3 . You missed out the 1/3.

Also, i understood the problem like this.

It is required to find the rate of ht change when t= 25.

so the vol when t =25 will be 8 pi * 25 = 400 pi.

Hence from 400 pi to the full we need to calculate the rate dh/dt. Is this correct?

so, i got V = 4/3 pi a^3 - pi * h^3/48

diff w .r. t ' h' and proceed to use the chain rule. dh/dt = dh/dV . dV/dt

The answer in the book was '0.011' cm/s

If the method is right then it is still alright.

[aldld]

HI

I did proceed in a similar way. But I reckon there is an error in equating the 'V'.

It must be * 1/3 . You missed out the 1/3.

Oh yeah, thanks I seem to have dropped out pi/3 by accident (I believe there should be a pi as well)