Work: Pumping a Tank — Calc II Explorer

Units

ft · lb m · kg

The tank

length = ft

width = ft

height H = ft

The first three stand on the ground with their floor at y = 0: a box and a cylinder have the same cross-section at every height, while the cone stands on its vertex, so its cross-section is a point at the bottom and a circle of radius R at the rim. The lying cylinder and the ball are measured from their middle instead — y = 0 is the centre line, so y runs from −R at the bottom to R at the top and the cross-section is widest in the middle.

The liquid

filled to h = ft

weight density ρ = lb/ft³

pump up to T = ft

T is measured from the tank's floor like everything else, so T = H pumps just over the rim and a larger T adds a spout above it. Water weighs about 62.5 lb/ft³; in metric you enter its mass density (1000 kg/m³) instead, which is why a g appears in the layer's weight beside it.

The tank

y =

How to set it up

Work out each step for yourself, then open it to check.

Every drop in one layer rises by the same amount, so the layer is a constant weight moved a constant distance — that is what makes the last step a sum you can integrate.