The height of a conical sand pile is always twice the radius.If sand is being added to the pile at a rate of ,how fast is the height of the pile increasing when the circumference of the base of the sand pile is ?

Question

The height of a conical sand pile is always twice the radius.If sand is being added to the pile at a rate of  ,how fast is the height of the pile increasing when the circumference of the base of the sand pile is  ?​ ​  ​​

Quizplus · Accepted Answer

Let the radius of the base be "r" and the height of the cone be "h". We know that h = 2r.
The formula for the volume of a cone is V = (1/3)*π*r^2*h.
Differentiating with respect to time, we get dV/dt = (1/3)*π*[2r*(dr/dt)*h + r^2*(dh/dt)].
We are given the value of dV/dt = 11.
We also know that the circumference of the base is 2πr = 11, which means r = 11/(2π).
Substituting the values in the equation, we get 11 = (1/3)*π*[2*(11/(2π))*((dr/dt)*2*(11/(2π))) + (11/(2π))^2*(dh/dt)].
Simplifying, we get dh/dt = (22/π^3)*(π - √3) ≈ 9.28.
Therefore, the height of the pile is increasing at a rate of approximately 9.28 units per unit time.

The Height of a Conical Sand Pile Is Always Twice