Math CentralQuandaries & Queries


Question from MJ, a student:

Suppose that grain pouring from a chute forms a conical heap in such a way that the height is always 2/3 the radius of the base. At the moment when the conical heap is 3 m high, its height is rising at the rate of 1/2 m/min. At what rate (in m^3/min) is the grain pouring from the chute?


Let's try a somewhat simpler problem first.

Water is being pumped into a spherical balloon at a constant rate. At the moment when the radius is 2 m, the radius is increasing at 1 m/hr. At what rate is the water being pumped into the balloon?

If $V(t)$ is the volume of the balloon in $m^3$ at time $t$ hours and $r(t)$ is the radius on the balloon in $m$ at time $t$ hours then

\[V(t) = \frac43 \pi \; r(t)^3.\]

You can differentiate this expression with respect to $t$ to obtain

\[V^{\;\prime}(t) = 4 \pi \; r(t)^2 r^\prime(t)\]

You are told that at some specific time $t^*$ the radius is 2 m and the radius is increasing at 1 m/hr. That is $r(t^*) = 2$ m and $ r^\prime(t^*) = 1$ m/hr. Thus at this time

\[V^{\;\prime}(t^*) = 4 \pi \; r(t^*)^2 r^\prime(t^*) = 4 \pi \times 2^2 \times 1 = 16 \pi\; m^3 \mbox{ per hour}\]

Since the water is being pumped into the balloon at a constant rate, $V(t)$ is a constant and hence $V(t) = 16 \pi\; m^3 \mbox{ per hour}$ for all $t.$

Your problem is quite similar. You have a conical shape and the volume $V(t)$ of a cone is given by

\[V(t) = \frac13 \pi \; r(t)^2 h(t)\]

where $r(t)$ is the radius and $h(t)$ is the height. The complication you have is that $V(t)$ is a function of two variables $r(t)$ and $h(t)$ and you only know values of $r(t)$ and $r^\prime(t)$ so somehow you need to eliminate $h(t)$ from the expression for the volume. Read the problem again and you see that "the height is always 2/3 the radius" and hence $h(t) = \frac23 r(t).$ Substitute this into the expression for the volume and you will then have the volume as a function of one variable $r(t).$ Now you can proceed as in my problem with the balloon.

I hope this helps,

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