Question

Spheres made of lead having radii of r, 2r, and 4r are melted to form a new sphere of radius R. The ratio of the volume to the surface area of the new sphere is equal to 4.18. Compute the value of 'r'.

Answer 1

Solution)

Given that spheres are made of radii, r,2r and 4r

New sphere has radius = R

Volume of lead in 3 spheres = V= (4/3)π(r³+(2r)³+(4r)³)

=>Vo= (4/3)πr³(1+8+64)= (292/3)πr³

Volume of new cube= Vn= (4/3)πR³

Surface area of new cube= S= 4πR²

But we have,

Vo= Vn (as volume is conserved)

=>(292/3)πr³= (4/3)πR³

=>73r³=R³ .....(I)

Also,

Vn/S= 4.18

=>{(4/3)πR³}/(4πR²)=4.18

=>R=4.18*3= 12.54,

Substituting in equation I and solving for r we get

r= 12.54/73⅓=3 units.....(answer)