Question

a machine starts dumping sand at the rate 20 m^3/min, forming a pile in the shape of a cone. The height of the pile is always twice the length of the base diameter. After 5 minutes, what is the height of the pile? Please include steps. Thanks!

Answer 1

11.52 meters.

Explanation:

At the rate of 20 m^3/min, a machine is dumping sand, forming a pile of the shape of a cone.

Therefore, in 5 minutes it will dump (20 × 5) = 100 m³.

Now, the length of the base diameter is always half of the height of the pile.

Let the height after 5 minutes is h meters.

So, h = 2(2r) = 4r {Where, r is the radius of the cone after 5 minutes} ........ (1)

Now, volume of a cone is given by
`V = (1)/(3) \pi r^(2) h`

⇒
`100 = (1)/(3) * ((22)/(7)) * ((h)/(4) )^(2) * h` {From equation (1)}

⇒
`h^(3) = 1527.27`

⇒ h = 11.52 meters. (Answer)