Calculating DH/dT for a Cone: Understanding dV/dH in the Volume Formula

[quicksilver123]

Both the problem and my attempt at a solution are provided. However, I become stuck.

The answer book (question 27, as pictured in the next post, the upload size limit made me create a second post and both images are about 4mb), suggests that I use dV/dH, which is the portion of the cone volume formula: pih^2/4 (once the square is distributed to the radius).

I am having trouble understanding why this part of the formula is considered to be dV/dH, and I am not especially strong in Leibniz notation; I prefer to use prime notation.
I suppose that weakness is catching up with me on this problem.

Thank you in advance for any replies.

 

[quicksilver123]

 

[TSny]

The first line of the solution of #27 shows that $$V = \frac{\pi h^3}{12}$$ Are you asking how they get from this to $$\frac{dV}{dh} = \frac{\pi h^2}{4} $$

 

[quicksilver123]

That is exactly what I'm asking, thank you.

Maybe it would help if you explained exactly what dV/dH meant? I think its something like the derivative of V with respect to H...
I get dV/dT meaning flow rate, which makes intuitive sense to me, but I don't understand how the expression shown above equals dV/dH

 

[TSny]

It's straight-forward differentiation. What is the result of $$\frac{d}{dh}(h^3)$$

 

[Mark44]

Maybe it would help if you explained exactly what dV/dH meant? I think its something like the derivative of V with respect to H...

dVdh is the rate of change of volume V with respect to height h.

In this problem, both V and h are functions of time, so differentiate your equation involving V and h with respect to t, using the chain rule.

I get dV/dT meaning flow rate, which makes intuitive sense to me, but I don't understand how the expression shown above equals dV/dH

 

[quicksilver123]

aie/// i don't get it

well, i get that its the cahnge of volume with respect to height but I'm not sure about how to apply that in this question, as in what expression explains it

 

[Mark44]

aie/// i don't get it

well, i get that its the cahnge of volume with respect to height but I'm not sure about how to apply that in this question, as in what expression explains it

You have $V = \frac{\pi h^3}{12}$
This is the relationship between V and h. In a related rates problem, you have to get the relationship between the rates (derivatives).

Differentiate both sides of the equation above with respect to t. That will give you dV/dt on the left side. You'll need to use the chain rule to get the derivative with respect to t of the other side.