Asked about: Volume of Cones

What is the base diameter of 1000 tons of stockpiled 1" rock?

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Brian Monetti, I spent several years working in home improvement and construction

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We will have to make some estimates here, but we can probably get a pretty accurate answer. Lets just assume the pile is a cone, it will make the math easier, and is probably pretty accurate. We will need the volume of rock. Crushed stone is somewhere around 100 lbs/cf. That means our pile has 20,000 cubic feet of stone (1000 tons x 2000 lbs per ton = 2000000 lbs, which divided by 100lbs/cf gives us 20,000 cubic feet.)

The volume of a cone = (pi) x (radius squared) x (height) / 3. We now know the volume, but need to find the radius and height. This is where it gets cool. Whenever you pile a material, it will always have a slope of the same angle. Civil Engineers call this the angle of repose. If the angle of a pile is too steep, it will basically landslide down until it is at the angle of repose. You can learn more about it here: http://en.wikipedia.org/wiki/Angle_of_repose

Now it turns out that 37 degrees is a pretty good estimate for crushed stone. That means the angle between the ground and the slope of the stone pile will be 37 degrees. Now imagine slicing the pile on half. You form a triangle between the bottom corner of the cone, the center, and the top of the pile. The bottom leg of the triangle is the Radius, and the height is the Height.

We have to use some trigonometry here to get the ratio between the height and radius. To do this, we use the Tangent. The tangent of an angle is equal to the opposite side length divided by the adjacent leg length. So, the ratio between the height of the triangle and the radius of cone (the leg thats on the ground) will be the Tangent of 37. Luckily for us, the tangent of 37 is very close to .75, so the ratio is 3 to 4 for height to radius of our cone to height.

So now we plug that into our volume equation for a cone. Volume (20,000) = (pi) x (radius squared) x (height) / 3. We know the ratio of height to radius is 3 to 4, so we can use a variable there and solve for the variable to find the radius. Basically, we set the Radius to equal 4n and the height to equal 3n, and then we can solve for n. So our equation will look like:

20,000 = (pi) x (4n)squared x (3n) / 3

Solving for n, we get 7.35. This means the radius is actually 4 times that because the radius is 4n. Therefore, the radius of the pile will be 7.35 x 4, or 29.4 feet, and the diameter of the pile will be 58 feet.

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