A hollow ball is to be made out of 27π cubic inch f rubber. If the ball will have uniform thickness, & desired surface area is 36π square inches, how thick should the material of this ball be?
Let ball radius (outer) = ro
" " " (inner) = ri
Then thickness, t = ro - ri
Vo = (4/3).pi.ro^3
Vi = (4/3).pi.ri^3
ΔV = Vo - Vi = (4/3).pi(ro^3 - ri^3)
But ΔV = 27.pi, So,
27.pi = (4/3).pi(ro^3 - ri^3)
(ro^3 - ri^3) = 51/4
Surface Area
Ao = 4.pi.ro^2 (assuming here that the surface area being referred to is that of the outer surface)
4.pi.ro^2 = 36.pi
ro^2 = 9
ro = 3
Using (ro^3 - ri^3) = 51/4
27 - ri^3 = 51/4
ri^3 = 27(4/4) - 3*27/4 = 27/4
ri = (3/2)³√(2) (1.25992...)
Thickness, t = ro - ri = 3 - (3/2)³√(2) = 3 - 1.88988... = 1.110...