Calculating the Speed of a Sand Pile's Height Increase

In summary, the problem involves a cone with a height that is always half the diameter of its base. The sand is falling at a rate of 1/2 cubic meters per minute. The goal is to find the rate of change of the height when the pile is 3 meters high. Using the equation V = (1/12)pih3, we can solve for dh/dt by setting h=3 and using r=4h instead of h/4.
  • #1
superjen
26
0
Sand is falling into a cone whose height is always 1/2 the diameter of the base. If the sand is falling at a rate of 1/2 cubic meters per min. How fast is the height increasin when the pile is 3 meters high?

so far I've drawn my diagram and got this far.

Given:
h = 1/2 the base
dv/dt = +1/2m3/min

Needs:
dh/dt

the equation i was going to use is
V = 1/12pih3

then i get

dv/dt = (1/2pi)(3h2)dh/dt

i don't know how to get the h?
any help?
 
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  • #2
You're asked to find dh/dt when the height is 3 meters...so h=3:wink:

But there is a problem with your starting equation V = (1/12)pih3...If the height is always half the diameter, then r=4h not h/4
 
  • #3
ok, that was stupid of me,
i don't even know why i was having trouble with that now >.<
 

FAQ: Calculating the Speed of a Sand Pile's Height Increase

What is the formula for calculating the speed of a sand pile's height increase?

The formula for calculating the speed of a sand pile's height increase is speed = volume / time. This means that the speed is equal to the change in volume over the change in time.

How do you measure the volume of a sand pile?

The volume of a sand pile can be measured by first measuring the length, width, and height of the pile. Then, using the formula volume = length x width x height, you can calculate the total volume of the sand pile.

How do you measure the time it takes for a sand pile to increase in height?

The time it takes for a sand pile to increase in height can be measured using a stopwatch. Start the stopwatch when the sand pile starts to increase in height and stop it once the desired height is reached. This will give you the total time it took for the sand pile to increase in height.

Can the speed of a sand pile's height increase be affected by other factors?

Yes, the speed of a sand pile's height increase can be affected by various factors such as the type of sand, the angle at which the sand is piled, and external forces such as wind or vibrations. These factors can impact the overall stability and rate of change of the sand pile.

How can calculating the speed of a sand pile's height increase be useful?

Calculating the speed of a sand pile's height increase can be useful in understanding the behavior and properties of sand piles. It can also be applied in engineering and construction to determine the stability and safety of structures built on top of or near sand piles. Additionally, it can also provide insights into the effects of external factors on the rate of change of sand piles.

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