Sand is being dumped from a dump truck at the rate of and forms a pile in the shape of a cone whose height is always half of its radius. How fast is its height rising when the pile is ?

Question

Sand is being dumped from a dump truck at the rate of  and forms a pile in the shape of a cone whose height is always half of its radius. How fast is its height rising when the pile is  ?

Yichen · Accepted Answer

The pile of sand forms a cone whose volume is
  
Where
 radius of the conical pile
 height of the conical pile
 
We can also derive the following relationship between  and  based from the given as follows.
  
Or

 
So that the volume formula can now be written in terms of  only.
   
  
 
Differentiating this with respect to the time  and applying the Chain rule, we get
  
 
With the following information given, as well
  
  
We can now solve for the unknown which gives us
    
 
Therefore, the height of the pile is rising at a rate of    .

Sand is being dumped from a dump truck at the rate of and forms a pile in the shape of a cone whose height is always half of its radius. How fast is its height rising when the pile is ?

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