Sand pouring from a chute forms a conical pile whose height is always equal to the diameter..
If the height increases at a rate of 5ft per/min, at what rate is the sand pouring from the chute when it is 10 ft high?
The rate of sand increasing height = 5ft/min
The time required to fill 10 ft height = 10/5 min = 2min
When the height of the sand is 10ft then the diameter of the sand is also 10ft.
Volume of a cone `= 1/3*pi*r^2*h` with standard notations.
Volume of sand filled `= 1/3*pi*(10/2)^2*10 ft^3 = 261.8 ft^3`
This volume is filled in 2 minutes.
Rate of the sand pouring from chute `= 261.8/2 ft^3/s = 130.9f^3/s`
So the rate of the sand pouring from chute is 130.9 cubic feet per second.