Question

Sand falls from an overhead bin and accumulates in a conical pile with a radius that is always threethree times its height. Suppose the height of the pile increases at a rate of 22 cm divided by scm/s when the pile is 1616 cm high. At what rate is the sand leaving the bin at that​ instant?

Answer 1

159241.048 cm³/s

Step-by-step explanation:

r = Radius = 3×height = 3h

h = height = 16 cm

Height of the pile increases at a rate =
`(dh)/(dt)=22\ cm/s`

`\text{Volume of cone}=(1)/(3)\pi r^2h\\\Rightarrow V=(1)/(3)\pi (3h)^2h\\\Rightarrow V=3\pi h^3`

Differentiating with respect to time

`(dv)/(dt)=9\pi h^2(dh)/(dt)\\\Rightarrow (dv)/(dt)=9\pi 16^2* 22\\\Rightarrow (dv)/(dt)=159241.048\ cm^3/s`

∴ Rate is the sand leaving the bin at that​ instant is 159241.048 cm³/s