I have a silo with the shape of a cylinder on top and a right cone below it. I know the radius and height of the cone and cylinder. When the sand is first loaded into the silo, the top is (mostly) flat. Then, as the sand is taken out of the silo (from the bottom), a cone of depression forms. However, once that cone forms, it stays uniform until the silo is empty. How do I figure out how much sand is in the silo if I just know the height of the sand in the cylinder, incorporating that cone of depression?
Assuming the apex of the cone points downward; the total capacity is the volume of the cylinder plus the volume of the cone ( πr^2h cyl + 1/3πr^2h cone). If the cones are not congruent, the amount of sand in the silo is total capacity minus the volume of the empty portion of the cylinder and the volume of the cone of depression. If the cones are congruent, the missing sand in the cone of depression will be equal to the sand at the base (as long as the apex of the cone of depression is above the base cone) and the volume of sand in the silo will be the volume of a cylinder with height determined by the distance between the bases. If the apex of the cone of depression falls beneath the base cone, the volume will approximately be the surface area of the come of depression × the distance between the apexes.
