Question

sand falls from an overhead bin and accumulates in a conical pile with a radius that is always four times its height. suppose the height of the pile increases at a rate of 3 cm/s when the pile is 20 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: Recall volume formula of cone

The volume formula of a cone is $V=\frac{1}{3}\pi r^{2}h$. Given that $r = 4h$, substitute $r$ into the volume formula: $V=\frac{1}{3}\pi(4h)^{2}h=\frac{16}{3}\pi h^{3}$.

Step2: Differentiate with respect to time $t$

Using the chain - rule, $\frac{dV}{dt}=\frac{16}{3}\pi\times3h^{2}\frac{dh}{dt}=16\pi h^{2}\frac{dh}{dt}$.

Step3: Substitute given values

We know that $h = 20$ cm and $\frac{dh}{dt}=3$ cm/s. Substitute these values into the derivative formula: $\frac{dV}{dt}=16\pi\times(20)^{2}\times3$.

Step4: Calculate the result

$\frac{dV}{dt}=16\pi\times400\times3 = 19200\pi$ $cm^{3}/s$.

Answer:

$19200\pi$ $cm^{3}/s$