Serway & Jewett — Physics and Measurement: Problem 34

A spherical shell has an outside radius of $2.60\text{ cm}$ and an inside radius of $a$. The shell wall has uniform thickness and is made of a material with density $4.70\text{ g/cm}^3$. The space inside the shell is filled with a liquid having a density of $1.23\text{ g/cm}^3$. (a) Find the mass $m$ of the sphere, including its contents, as a function of $a$. (b) For what value of the variable $a$ does $m$ have its maximum possible value? (c) What is this maximum mass? (d) Explain whether the value from part (c) agrees with the result of a direct calculation of the mass of a solid sphere of uniform density made of the same material as the shell. (e) What If? Would the answer to part (a) change if the inner wall were not concentric with the outer wall?