To begin, we must determine the dimensions of the cube. To do this, recall that the surface area of a cube is made up of six squares and is thus defined as: A = 6s², where s is one of the sides of the cube. For our data, this gives us:

726 = 6s²; 121 = s²; s = 11

Now, if the sphere is contained within the cube, that means that 11 represents the diameter of the sphere. Therefore, the radius of the sphere is 5.5 units. The surface area of a sphere is defined as: A = 4πr². For our data, that would be:

A = 4π * 5.5² = 30.25 * 4 * π = 121π

The surface area of a sphere is 4πr². The area of a circle is πr². 16/16 is equal to 1.

Let's call the radius of the sphere r. The formula for the surface area of a sphere (A) is given below:

A = 4πr²

Because the sphere is inscribed inside the cube, the diameter of the sphere is equal to the side length of the cube. Because the diameter is twice the length of the radius, the diameter of the sphere is 2r. This means that the side length of the cube is also 2r. 

The surface area for a cube is given by the following formula, where s represents the length of each side of the cube:

surface area of cube = 6s²

The formula for surface area of a cube comes from the fact that each face of the cube has an area of s², and there are 6 faces total on a cube. 

Since we already determined that the side length of the cube is the same as 2r, we can replace s with 2r.

surface area of cube = 6(2r)² = 6(2r)(2r) = 24r².

We are asked to find the ratio of the surface area of the cube to the surface area of the sphere. This means we must divide the surface area of the cube by the surface area of the sphere.

ratio = (24r²)/(4πr²)

The r² term cancels in the numerator and denominator. Also, 24/4 simplifes to 6.

ratio = (24r²)/(4πr²) = 6/π

The answer is 6/π.

A hemisphere is half of a sphere.  The surface area is broken into two parts:  the spherical part and the circular base. 

The surface area of a sphere is given by .

So the surface area of the spherical part of a hemisphere is . 

The area of the circular base is given by .  The radius to use is half the diameter, or 2 cm.

The surface area of a sphere with radius  can be determined using the formula

.

The smallest sphere, with radius , has surface area 

The second-smallest sphere, with radius , has surface area 

The surface areas are in an arithmetic sequence; their common difference is the difference of these two surface areas, or

Since the six surface areas are in an arithmetic sequence, the surface area of the largest of the six spheres - that is, the sixth-smallest sphere - is

The radii of the spheres form an arithmetic sequence, with 

 and 

The common difference  can be computed as follows:

The second-smallest sphere has radius 

The surface area of a sphere with radius  can be determined using the formula

.

Setting , we get

.

To solve, use the formula for the surface are a of a sphere.

Substitute in the radius of one into the following equation.

Thus,

To solve, simply use the formula for the surface area of a sphere. Thus,

To solve, simply use the formula for the surface area of a sphere. Thus,

The surface area for a sphere is one of those formulas you are going to have to memorize. There isn't exactly an easy wasy to derive it. My only trick for differentiating it from other circular formulas is the fact that area is two-dimensional, so you only square the r, not cube it.

The surface area of a given sphere is represented by the equation

Substituting in our given surface area, we can simplify this equation and solve for r.