The volume of a cube is found by V = s³.  Since V = 64, s = 4.  The side of the cube is the same as the diameter of the sphere.  Since d = 4, r = 2.  The surface area of a sphere is found by SA = 4π(r²) = 4π(2²) = 16π.

The surface area of the sphere before it was cut is equal to the following:

surface area of solid sphere = 4πr², where r is the length of the radius.

Each hemisphere will have the following shape:

In order to determine the surface area of the hemisphere, we must find the surface area of the flat region and the curved region. The flat region will have a surface area equal to the area of a circle with radius r.

area of flat part of hemisphere = πr²

The surface area of the curved portion of the hemisphere will equal one-half of the surface area of the uncut sphere, which we established to be 4πr².

area of curved part of hemisphere = (1/2)4πr² = 2πr²

The total surface area of the hemisphere will be equal to the sum of the surface areas of the flat part and curved part of the hemisphere.

total surface area of hemisphere = πr² + 2πr² = 3πr²

Finally, we must find the ratio of the surface area of the hemisphere to the surface area of the uncut sphere.

ratio = (3πr²)/(4πr²) = 3/4

The answer is 3/4.

To solve this, we must first begin by finding the radius of the sphere. To do this, recall that the volume of a sphere is:

V = (4/3)πr³

For our data, we can say:

2304π = (4/3)πr³; 2304 = (4/3)r³; 6912 = 4r³; 1728 = r³; 12 * 12 * 12 = r³; r = 12

Now, based on this, we can ascertain the surface area using the equation:

A = 4πr²

For our data, this is:

A = 4π*12² = 576π

To find the surface area, we must first find the radius.  Based on our description, this passes from (0,0) to (6,8).  This can be found using the distance formula:

6² + 8² = r²; r² = 36 + 64; r² = 100; r = 10

It should be noted that you could have quickly figured this out by seeing that (6,8) is the hypotenuse of a 6-8-10 triangle (which is a multiple of the "easy" 3-4-5).

The rest is easy.  The surface area of the sphere is defined by:

A = 4πr² = 4 * 100 * π = 400π

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

.