Question

A spherical shell has inner radius r and an outer radius r+h. Let h=0.1r. Find the volume of the shell in terms of r and then approximate the volume of the shell by 4πr^2h. Does this seem like a reasonable approximation if h is small compared to r?

Answer 1

Explanation:

The volume of the shell is the volume of the outer sphere minus the volume of the inner sphere.

V = 4/3 π (r+h)³ − 4/3 π r³

V = 4/3 π (r³ + 3r²h + 3rh² + h³) − 4/3 π r³

V = 4/3 π r³ + 4π r²h + 4π rh² + 4/3 π h³ − 4/3 π r³

V = 4π r²h + 4π rh² + 4/3 π h³

If h = 0.1 r:

V = 4π r² (0.1 r) + 4π r (0.1 r)² + 4/3 π (0.1 r)³

V = 0.4π r³ + 0.04π r³ + 0.00133π r³

V = 0.44133π r³

Using the approximation:

V = 4π r² h

V = 4π r² (0.1 r)

V = 0.4π r³

The approximation is reasonable if h is small compared to r.