The surface area of a cylinder is calculated as a top, a bottom, and the outside. The outside is measured as the circumference * height, or . That will be the same size as the label. Therefore, . We know that , therefore, , and .

The volume of the cylinder is calculated as . Make sure that you square the full thing, . You will end up with .

The volume of a cube is  (that's why you call  to the third power "-cubed"), where  represents the length of one of the sides. The surface area is , or more conceptually the sum of the areas of each of the six faces (top, bottom, front, back, left, right) of the cube.

So if the volume of a cube is , that means that , so .

Plugging that into the surface area formula, you have  which is .

When looking to find the relationship between the volumes of the two spherical figures in the question stem, we’ll want to keep in mind the formula for the volume of a sphere:

So, if we plug each radius into this formula, we’ll arrive at:

Thus, the ratio between the two volumes is: 

Since the  component cancels in both the numerator and the denominator, the ratio of the volumes is 1:64. Be sure to arrange the ratio in the order asked for in the question stem! Additionally, you’ll want to keep in mind that when a change in scale is applied to multiple dimensions, the scale changes exponentially! We can actually save some time on questions like this if we recognize this relationship.  

For instance, here, the “scale factor,” or change in scale from the smaller radius to the larger radius is 4, but since the relationship we’re looking for compares volume to volume, we need to account for the fact that the scale factor has been taken to the third power for the three dimensional measurement at hand. Thus, the ratio of the volumes will be 1 to 4³, or 1 to 64.

First we need to convert degrees to radians by multiplying by :

Now we can write: 

First, we need to convert  to radians by multiplying by : 

Now we can solve the following equation for :

Since , we can solve by setting up a proportion:

Cross multiply and solve.

First, we need to simplify the expression:

Now multiply by :

First, we need to simplify the expression:

First, we need to simplify the expression:

Then multiply by :

Recall the definition of "radians" derived from the unit circle:

The quantity of radians given in the problem is . All that is required to convert this measure into degrees is to denote the unknown angle measure in degrees by  and set up a proportion equation using the aforementioned definition relating radians to degrees:

Cross-multiply the denominators in these fractions to obtain:

or

.

Canceling like terms in these equations yields

Hence, the correct angle measure of  in degrees is .