tricky differentiation question help

"a hollow right circular cone has a height 18cm and base radius 12cm. It is held vertex downwards beneath a tap leaking at a rate of 2cm^3s-1. Find the rate of rise of water level when the depth is 6cm.

Ok so

dv/dt=2

and we have to find dh/dt (h=height)

so dv/dt = dv/dh x dh/dt

to find dv/dh i found the radius of the cone a 6cm from the vertex by using direct proportion and i got it to be 4cm. I then used v=1/3 pi r^2 h and substituted my new value of r (4cm) and differentiated to get dv/dh= (16/3)pi

I then did 2=(16/3)pi x dh/dt and rearranged to find dh/dt which is the rate of change of the height of the water when the radius=4 i.e the height=6... but i dont get the right answer


is my method wrong? thanks

I think so. You need V in terms of h only before finding dV/dh
so you should be differentiating $V= \frac{4\pi}{27}h^3$

do you not just differentate the equation for the volume of a cone and insert the value of the radius? where do you get that equation?

You can only differentiate something if it is a function of a single variable, so use the constraints you have been given in the question

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please explain....im so lost

Use the constraints given in the question to express the volume of the cone in terms of it's height (Hint: use the formula for volume of a cone)

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but the volume of a cone is $\frac{1}{3}\pi r^2 h$ ?

And you need to find a way to get that formula in terms of a single variable (in this case h) before you can differentiate it.

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why can we just h=6?

You need to differentiate first, then you can substitute in the h=6, or equivalant, into the derivative. Although you're evaluating the derivative at a specific value of h, h is not a constant in the formula, it varies as r varies.

I think so. You need V in terms of h only before finding dV/dh
so you should be differentiating $V= \frac{4\pi}{27}h^3$

ok so i guess i use Pythagoras theorem to find the slope of the cone the sub r^2 into the formula for the volume?

Don't know what Pythagoras has to do with it. It should be clear that the r is proportional to h, and since you know r=12 when h=18, the relationship between the two is....

yea r = 4 when h =6?

how exactly do you get this? iv tried everything and i cannot get it!