Rates of change

**requal** · July 12th 2009, 11:16 PM

Sand being poured from a conveyor belt forms a cone with height h and a semi-vertical angle of 60 degrees. Let the volume of the sand is $\pi h^3$ .

Suppose that the sand is being poured at a constant rate of 0.3m^3/min. Find the rate of increase of the base area.

**pickslides** · July 12th 2009, 11:36 PM

I can suggest

$\frac{dV}{dh} = 3\pi h^2$

$\frac{dV}{dt} = 0.3$

$\frac{dA}{dr} = 2\pi r$

**HallsofIvy** · July 13th 2009, 06:19 AM

Was $V= \pi h^3$ given in the problem?
The volume of a cone, of height h and base radius r, is given by $V= \frac{1}{3}\pi r^2 h$ . Here, you are told that the "semi-vertical angle" is 60 degrees so using $\frac{r}{h}= tan(60)= \sqrt{3}$ . $r= \sqrt{3}h$ , $r^2= 3h^2$ and $V= \pi h^3$ . That is where that formula came from.

To change to a "rate of change" formula, differentiate both sides with respect to t, using the chain rule.

$\frac{dV}{dt}= \frac{d\left(\pi h^3\right)}{dt}= 3\pi h^2 \frac{dh}{dt}$
Using the fact that $\frac{dV}{dt}= .3$ , you can solve that for $\frac{dh}{dt}$ , as a function of h.

The area of the base is, of course, $A= \pi r^2= 3\pi h^2$ so $\frac{dA}{dt}= 6\pi h\frac{dh}{dt}$ .

**requal** · July 14th 2009, 09:17 PM

Originally Posted by HallsofIvy

Was $V= \pi h^3$ given in the problem?
The volume of a cone, of height h and base radius r, is given by $V= \frac{1}{3}\pi r^2 h$ . Here, you are told that the "semi-vertical angle" is 60 degrees so using $\frac{r}{h}= tan(60)= \sqrt{3}$ . $r= \sqrt{3}h$ , $r^2= 3h^2 and$ V= \pi h^3[/tex]. That is where that formula came from.

To change to a "rate of change" formula, differentiate both sides with respect to t, using the chain rule.

$\frac{dV}{dt}= \frac{d\left(\pi h^3\right)}{dt}= 3\pi h^2 \frac{dh}{dt}$
Using the fact that $\frac{dV}{dt}= .3$ , you can solve that for $\frac{dh}{dt}$ , as a function of h.

The area of the base is, of course, $A= \pi r^2= 3\pi h^2$ so $\frac{dA}{dt}= 6\pi h\frac{dh}{dt}$ .

Yes but then, what- I get a figure which still has the pronumeral "h"

**Grandad** · July 15th 2009, 01:48 AM

Hello requal

Originally Posted by requal

Yes but then, what- I get a figure which still has the pronumeral "h"

Unless you are given a specific value of $h$ , that's all you will be able to do: express the rate of change of $A$ as a function of $h$ .

Grandad

**requal** · July 15th 2009, 07:21 PM

just wondering, can't I just diffentiate the area in terms of volume(da/dv) so I can use da/dt=da/dv*dv/dt?

**Grandad** · July 15th 2009, 09:53 PM

Hello requal

Originally Posted by requal

just wondering, can't I just diffentiate the area in terms of volume(da/dv) so I can use da/dt=da/dv*dv/dt?

You could do, by eliminating $h$ between the equations $V= \pi h^3$ and $A = 3\pi h^2$ , but you'd then end up with an expression for $\frac{dA}{dt}$ in terms of $V$ (or $A$ , or both).

Grandad

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