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[SOLVED] Volume of a cone change of rate of volume with respect to h and r

karush

Well-known member
Jan 31, 2012
2,771
(a) Find the rate of change of the volume with respect to the height if the radius is constant

vol of right circular cone is $$V=\frac{1}{3} \pi r^2 h$$

from this $$h=\frac{3V}{\pi r^2}$$

$$\frac{dh}{dt}=\frac{3}{\pi r}\frac{dV}{dt}$$

$$\frac{\pi r}{3}\frac{dh}{dt}=\frac{dV}{dt}$$

not sure about this we don't have t or rate of change of height:confused:

this next question is the same except height is constant

(b) Find the rate of change of the volume with respect to the radius if the height is constant.
 

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,140
Re: Volume of a cone change of rate of volumn in respect to h and r

(a) Find the rate of change of the volume with respect to the height if the radius is constant

vol of right circular cone is $$V=\frac{1}{3} \pi r^2 h$$

from this $$h=\frac{3V}{\pi r^2}$$

$$\frac{dh}{dt}=\frac{3}{\pi r}\frac{dV}{dt}$$

$$\frac{\pi r}{3}\frac{dh}{dt}=\frac{dV}{dt}$$

not sure about this we don't have t or rate of change of height:confused:

this next question is the same except height is constant

(b) Find the rate of change of the volume with respect to the radius if the height is constant.
They are looking for expressions for dh/dt and dr/dt in terms of variables. So solve your equation in a) for dh/dt.

-Dan
 

karush

Well-known member
Jan 31, 2012
2,771
Re: Volume of a cone change of rate of volumn in respect to h and r

so this is the ans for (a)

$$\frac{dh}{dt}=\frac{3}{\pi r}\frac{dV}{dt}$$
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Re: Volume of a cone change of rate of volumn in respect to h and r

The way I interpret these problems, there is no need to introduce a variable for time. You simply need to differentiate with respect to the stated variable.
 

karush

Well-known member
Jan 31, 2012
2,771
Re: Volume of a cone change of rate of volumn in respect to h and r

so just took out dt...

$$dh=\frac{3}{\pi r}dV$$
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Re: Volume of a cone change of rate of volumn in respect to h and r

(a) Find the rate of change of the volume with respect to the height if the radius is constant

(b) Find the rate of change of the volume with respect to the radius if the height is constant.
a) You are being asked to find \(\displaystyle \frac{dV}{dh}\).

b) You are being asked to find \(\displaystyle \frac{dV}{dr}\).
 

karush

Well-known member
Jan 31, 2012
2,771
Re: Volume of a cone change of rate of volumn in respect to h and r

$$ \displaystyle
dh=\frac{3}{\pi r}dV
\text { then }
\frac{dV}{dh}
=\frac{\pi\text{ r}}{3}
$$
$$\text{ and }$$
$$r=\sqrt{\frac{3V}{\pi h}}$$
$$\text { so }$$
$$dr=\frac{\sqrt{3}}{2h\sqrt{\frac{\pi v}{h}}}dV$$
$$\text{ and }$$
$$\frac{dV}{dr}=\frac{2h\sqrt{\frac{\pi v}{h}}}{\sqrt{3}}$$

I was expecting something more simple for answer?
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Re: Volume of a cone change of rate of volumn in respect to h and r

What I meant to do is as follows:

Given:

\(\displaystyle V=\frac{\pi}{3}r^2h\)

then:

\(\displaystyle \frac{dV}{dh}=\frac{\pi}{3}r^2\)

\(\displaystyle \frac{dV}{dr}=\frac{2\pi}{3}rh\)