A concrete spherical shell is constructed for building aesthetic. The contractor designated 6 in for its thickness and 24π sq ft for interior area, how much concrete is required for this project?
Let shell radius (outer) = ro
" " " (inner) = ri
Then thickness, t = ro – ri = 6 in
Volume (of shell)
Vo = (4/3).pi.ro^3
Vi = (4/3).pi.ri^3
ΔV = Vo - Vi = (4/3).pi(ro^3 - ri^3)
Surface Area
Ai = 4.pi.ri^2
4.pi.ri^2 = 24.pi*(12^2) square inches
ri^2 = 6*144
ri = 12√(6)
Using ro – ri = 6
ro = 6 + 12√(6)
ΔV = (4/3).pi(ro^3 - ri^3)
ΔV = (4/3).pi((6 + 12√(6) )^3 – (12√(6) )^3)
ΔV = (4/3).pi((35.394)^3 – (29.394)^3)
ΔV = (4/3).pi(18,942.54)
ΔV = 79,346.32 cubic inches = 45.92 cu ft