The area of the cross section of a sphere at the largest point is 100 pi square miles. what is the total surface area of the sphere?

1.
The cross-section of largest area is obtained when the cross-sectional plane, contains the center of the sphere.

That is, when this plane divides the sphere into 2 semi (half) spheres.

2.
This cross sectional region is a circle with surface area = 100 Pi square miles.

The area of a circle is , where R is the radius.

So

square miles,

3.

The surface area of a sphere is given by the formula 

So the surface area of this sphere is  square miles.

  square miles