Acube of edge length s is inscribed in a sphere of radius r. write the surface areasurface area upper ff of a cubecube as a function of the radius r. (b) find the instantaneous rate of change of the surface areasurface area upper ff with respect to radius r of the sphere sphere. (c) evaluate the rate of change of upper ff at r equals 2r=2 and r equals 8r=8. (d) if r is measured in millimeters millimeters and upper ff is measured innbsp square nbsp square millimeters, what units

Given that the length of the cube's edges is , then the length of the diagonal along any face of the cube is, by the Pythagorean theorem,



Then the diagonal along the opposite vertices of the cube (i.e. the line segment through the center of the cube joining its opposite corners) is, again by the Pythagorean theorem,



The radius of the sphere, , corresponds to half the length of the cube's diagonal; that is,



The surface area of the cube in terms of its edge length is



which means the area, rewritten in terms of the sphere's radius , is



Hence the rate of change of the cube's surface area with respect to a change in the sphere's radius is



When , we have



while when , we get



I'm not sure what part (d) reads, so I'll leave that to you...

Sorry can't answer that question