Question

Suppose the ends of the cylindrical storage tank in the figure arc circles of radius 3 ft and the cylinder is 20 ft long. Determine the volume of the oil in the tank to the nearest cubic foot if the rod shows a depth of 2 ft.

Answer 1

assuming the rod lies along a diameter of the circle, you need to find the area of a circular segment, knowing its height and the circle's radius.

Draw a diagram. If the depth is d, then the central angle subtending the surface of the oil can be found using

cos(θ/2) = (r-d)/r

and the area of the segment is

a = 1/2 r^2 (θ-sinθ)

Now multiply by the length of the tank to get the volume.

In one line, that would be

v = 20 (1/2 r^2 (θ-sinθ))
= 10*9 (2 arccos((r-d)/r) - √(r^2-(r-d)^2)
= 90(2 arccos(1/3) - √8/3)
= 136.72