# LessonRelated Rates: A Sand Pile

#### MarkFL

##### La Villa Strangiato
Staff member
Moderator
Math Helper
Sand is being dumped from a conveyor belt at a constant rate, forming a conical pile. The ratio of the base radius of this pile to its height is constant. Find the height of the pile as a function of time.

To begin this problem, let's define some constants:

$$a$$ = the rate at which sand is being dumped on the pile, and will have units of volume per time.

$$b$$ = the ratio of the base radius to the height of the pile.

And our variables:

$$V$$ = the volume of the pile at time $$t$$.

$$r$$ = the base radius of the pile.

$$h$$ = the height of the pile.

And so immediately, we may write:

$$\displaystyle \d{V}{t}=a$$

Now, the volume $$V$$ of a cone is given by:

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

We know:

$$\displaystyle b=\frac{r}{h}\implies r=bh$$

Hence:

$$\displaystyle V=\frac{\pi}{3}(bh)^2h=\frac{\pi b^2}{3}h^3$$

Implicitly differentiating with respect to $$t$$, we obtain:

$$\displaystyle \d{V}{t}=\pi b^2h^2\d{h}{t}$$

And so we have:

$$\displaystyle \frac{a}{\pi b^2}=h^2\d{h}{t}$$

Letting $$h_0$$ be the initial height of the pile, and exchanging dummy variables of integration, we may integrate as follows:

$$\displaystyle \frac{a}{\pi b^2}\int_0^t \,du=\int_{h_0}^{h} v^2\,dv$$

Applying the FTOC, there results:

$$\displaystyle \frac{a}{\pi b^2}t=\frac{1}{3}\left(h^3-h_0^3\right)$$

And so, we find:

$$\displaystyle h(t)=\sqrt{\frac{3a}{\pi b^2}t+h_0^3}$$

• harpazo and anemone

#### harpazo

##### Pure Mathematics
Is this in any way related to differential equations?

#### MarkFL

##### La Villa Strangiato
Staff member
Moderator
Math Helper
Is this in any way related to differential equations?
Yes, we wind up with an IVP in this problem, which we solve via integration.

• anemone and harpazo

#### harpazo

##### Pure Mathematics
Yes, we wind up with an IVP in this problem, which we solve via integration.
I guess that related rates is also taught in a differential equations course.

#### MarkFL

##### La Villa Strangiato
Staff member
I'm not really worried about what course this problem would be best taught in...it's related rates and the calculus can be used to solve it. • 