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?

Question

This is from my solid mensuration problem set

Answer 1

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...