The above diagram shows the situation.
To compute the volume of just the shell we have to calculate the volume of the bigger sphere, which has a radius of 4.5 cm, and then subtract to it the volume of the smaller sphere, which has a radius of 4 cm.
The volume of a sphere is computed as follows:
Substituting with r = 4.5 cm, the volume of the bigger sphere is:
![\begin{gathered} V_1=(4)/(3)\pi(4.5)^3 \\ V_1\approx381.7\operatorname{cm}^3 \end{gathered}]()
Substituting with r = 4 cm, the volume of the smaller sphere is:
![\begin{gathered} V_2=(4)/(3)\pi(4)^3 \\ V_2=268.08\operatorname{cm}^3 \end{gathered}]()
Finally, the volume of the shell is:
![\begin{gathered} V_{\text{shell}}=V_1-V_2 \\ V_{\text{shell}}=381.7-268.08 \\ V_{\text{shell}}=113.62\operatorname{cm}^3 \end{gathered}]()