Since the volume of the smaller cubes with edges, , is 27, we have:

.

The large cube has edges .

So the surface area of the large cube is:

.

The relationship can be determined, because it is possible to find the surface area of a cube from the volume and vice versa.

Quantity A:

To find the surface area of the cube, you must find the side length. To find the side length from the volume, you must find the cube root.  

Find the cube root of the volume.

Insert into surface area equation.

Quantity B:

To find the volume of a cube, you must find the side length. To find the side length from the surface area, you must divide by 6, then find the square root of the result. Then, cube that result.

Divide by 6.

Square root.

Now, to find the volume.

Quantity B is greater.

Begin by solving for the length of one side of the cube. Use the formula for surface area to do this:

s= length of one side of the cube

The length of the side of the cube is equal to the diameter of the sphere. Therefore, the radius of the sphere is 3. Now use the formula for the surface area of a sphere:

The surface area of the sphere is .

The surface area of a sphere is defined by the equation:

For our data, this means:

Solving for , we get:

 or 

The diameter of the sphere is .

Do not assume that the diameter will be half of the diameter of a sphere with volume of . Instead, begin with the sphere with a volume of . Such a simple action will prevent a vexing error!

Thus, we know from our equation for the volume of a sphere that:

Solving for , we get:

If you take the cube-root of both sides, you have:

First, you can factor out an :

Next, factor the :

Which simplifies to:

Thus, the diameter is double that or:

Surface Area = 4πr² = 4 * 22/7 * 7² = 616

The surface area of the larger sphere is NOT merely doubled from the smaller sphere, so we cannot double to find the answer.

We can use the surface area formula to find the radius of the original sphere.

r² = 4

r = 2

Therefore the larger sphere has a radius of 2 * 2 = 4.

The new surface area is then  square inches.

The volume of a cube is equal to .

So we mutiply our volume by  and divide by , giving us .

The surface area of a sphere is equal to , giving us .

Recall the equation for the volume of a sphere:

V = (4/3)πr³

If we increase the radius by 50%, we can represent the new radius as being equal to r + 0.5r = 1.5r.

Replace this into the equation for the volume and simplify:

V₂ = (4/3)π(1.5r)³ = (4/3)π(3.375r³)

Rewrite this so that you can compare the two volumes:

V₂ = 3.375 * (4/3)πr³ = 3.375 * [(4/3)πr³]

This is the same as:

V₂ = 3.375 * V

This means that the new volume is 337.5% of the original.  However, note that the question asked for the increase, which would be an increase by 237.5%.

Remember that the weight of an object is analogous to the volume. Since the weight of the sphere is 216, the volume of the sphere is proportional to 216. Remember the equation for volume of a sphere:

V = a * a * a = 216

Take the cube root of 216 to find that the length of one of the cubes is proportional to 6. According to the question, one of the sides of the cube is equivalent to the diameter of the sphere. 

Thus d = 6 and r = d/2 = 3 for the sphere. 

Remember the volume equation for a sphere:

V = 4/3 * π * r³

Plug in r = 3 to find V = 36π