Given data:
The diameter of the cut sphere, D=14 in.
The radius of the cut sphere is,
[tex]\begin{gathered} r=\frac{D}{2} \\ r=\frac{14}{2} \\ r=7\text{ in} \end{gathered}[/tex]The cut sphere is called a hemisphere.
The surface area of a sphere is
[tex]A_1=4\pi r^2[/tex]So, the lateral surface area of a hemisphere is half the surface area of sphere. Therefore, the lateral surface area of a hemisphere is,
[tex]\begin{gathered} A_2=\frac{4\pi r^2}{2} \\ A_2=2\pi r^2 \end{gathered}[/tex]The hemisphere has a lateral surface and a circular surface. The area of the circular surface is,
[tex]A_3=\pi r^2[/tex]Therefore, the total area of the hemisphere is,
[tex]\begin{gathered} A=A_2+A_3 \\ A=2\text{ }\pi r^2+\pi r^2 \\ A=3\text{ }\pi r^2 \end{gathered}[/tex]The total surface area of a hemisphere is,
[tex]\begin{gathered} A_{}=3\text{ }\pi r^2 \\ A=3\text{ }\pi\times7^2 \\ A=461.8in^2 \end{gathered}[/tex]Therefore, the total surface area of the cut sphere is 461.8 square inches.