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 11 cm high. at what rate is the sand leaving the bin at that instant?\nthe sand is leaving the bin at a rate of \n(type an exact answer, using \\( \\pi \\) 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 11 cm high. at what rate is the sand leaving the bin at that instant?\nthe sand is leaving the bin at a rate of \n(type an exact answer, using \\( \\pi \\) as needed.)

Answer

Explanation:

Step1: Write the volume formula for a cone

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

Step2: Differentiate the volume formula with respect to time (t)

Using the chain rule (\frac{dV}{dt}=\frac{dV}{dh}\cdot\frac{dh}{dt}). Differentiate (V=\frac{16}{3}\pi h^{3}) with respect to (h): (\frac{dV}{dh}=16\pi h^{2}). We know that (\frac{dh}{dt} = 1\space cm/s).

Step3: Substitute the value of (h)

When (h = 11\space cm), (\frac{dV}{dt}=16\pi(11)^{2}\cdot1). Calculate (16\pi\times121=1936\pi\space cm^{3}/s).

Answer:

(1936\pi)