Question

sand falls from an overhead bin and accumulates in a conical pile with a radius that is always three times its height. suppose the height of the pile increases at a rate of 1 cm/s when the pile is 16 cm high. at what rate is the sand leaving the bin at that instant? the sand is leaving the bin at a rate of (type an exact answer, using π as needed.)

Explanation:

Step1: Write the volume formula for a cone

The volume formula of a cone is $V=\frac{1}{3}\pi r^{2}h$. Given that $r = 3h$, we substitute $r$ into the volume formula: $V=\frac{1}{3}\pi(3h)^{2}h=\frac{1}{3}\pi\times9h^{2}\times h = 3\pi h^{3}$.

Step2: Differentiate the volume with respect to time

Differentiate $V = 3\pi h^{3}$ with respect to time $t$ using the chain - rule. $\frac{dV}{dt}=9\pi h^{2}\frac{dh}{dt}$.

Step3: Substitute the given values

We are given that $\frac{dh}{dt}=1$ cm/s and $h = 16$ cm. Substitute these values into the derivative formula: $\frac{dV}{dt}=9\pi\times(16)^{2}\times1$.
Calculate $9\pi\times256 = 2304\pi$ $cm^{3}/s$.

Answer:

$2304\pi$ $cm^{3}/s$