sand falls from an overhead bin and accumulates in a conical pile with a radius that is always four times…

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 1 cm/s when the pile is 10 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.)

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 1 cm/s when the pile is 10 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.)

Answer

Explanation:

Step1: Write volume formula for cone

The volume formula for a cone is $V=\frac{1}{3}\pi r^{2}h$. Given that $r = 4h$, we 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

Differentiate $V$ with respect to time $t$ using the chain - rule. $\frac{dV}{dt}=\frac{dV}{dh}\cdot\frac{dh}{dt}$. First, find $\frac{dV}{dh}$: $\frac{dV}{dh}=\frac{16}{3}\pi\cdot3h^{2}=16\pi h^{2}$. Then, since $\frac{dh}{dt}=1$ cm/s.

Step3: Substitute the value of h

We know that $h = 10$ cm. Substitute $h = 10$ into $\frac{dV}{dt}=16\pi h^{2}\cdot\frac{dh}{dt}$. So $\frac{dV}{dt}=16\pi(10)^{2}\cdot1$.

Answer:

$1600\pi$ cm³/s