The relationships between side lengths and area for two dimensional figures, and side lengths and volume for three dimensional figures can be confusing. It is very easy to think that if the lengths all double, the area and volume should follow suit, but if every dimension is doubled in a 2-dimensional shape (a square or rectangle, where you're talking about area) the area is multiplied by  (the square of the change for the sides) and in a 3-dimensional shape (a cube, sphere, or box, where you're talking about volume) the volume is multiplied by 8 (the cube of the change of the sides).

Suppose that the length of each side of the smaller cube is . That would mean that the volume would be  If you double that length across all dimensions, then in calculating the length, width, and depth you'd multiply: 

This simplifies to . And note the relationship between the larger and smaller cubes: , the volume of the larger cube, . So with the given problem, if the volume of the smaller cube is , then the volume of the larger cube is 8 times that, so the correct answer is .

This problem asks you to find the surface area for  sides of the box, since the top side will not have area. You should then determine the dimensions of each side that you'll be using.

For the left and right sides, the measurement will be  square inches, and since you'll have two of those sides you'll multiply by  to have  square inches of sides.

The front and back will measure  square inches, and since you'll have two of those sides you should multiply by  to have  square inches of front/back.

Then you'll need to account for the bottom, which measures  square inches.

So your total calculation is  square inches.

The surface area of a cylinder has three components: the area of the top, the area of the bottom, and the area of the "side."

The areas of the top and bottom are classic circles, so you'll use  to calculate. Here the radius is , so each circle will have a volume of . One important key here is to remember to multiply that by 2 to account for both circles. So combined, the top and bottom have an area of .

For the "side," it is important to think conceptually about what constitutes that area. If you were to unroll the circular nature of the cylinder, the side would form a rectangle. Quite clearly the height will be the same as the height of the cylinder, but what about the length? The length is the circumference of the circle, the distance along the top (or bottom) for the material to stretch exactly around the circle.

Circumference is , so here that's . Multiply that by the height of  and you have  as the area of the side. So your area is now , which sums to .

Importantly here, the greatest distance between two points in the cube (from one corner to the opposite corner) will equal the diameter of the sphere. Because the cube is perfectly inscribed within the circle, a line that travels through the center of the cube will travel through the center of the sphere, and if it touches two corners of the cube then it's touching the outside of the sphere, satisfying the definition of the diameter.

With that, your goal should be to use the volume of the cube to determine the diameter of the sphere. This can be done quickly if you know the rule for the greatest distance in a rectangular box: . Here since length, width, and height are all the same, , you have a quick calculation:

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Since the circumference of a circle can be expressed as , your answer is simply .

The volume of a rectangular box is Length × Width × Height. Here you're given those three dimensions as , but then told that the volume of water is only of the total. So your calculation is , which comes out to .

An important lesson from this problem involves the relationship between length and volume. Since volume is three-dimensional and length is only one-dimensional, when you reduce the length of all sides in a 3-D shape, you have to account for that change along all three dimensions. In this case, the box is scaled down by a linear factor of , so its volume scales down by a factor of .

So the new box is  the volume of the old box, meaning that it decreased in volume by .  expressed as a percentage is .

Alternatively, you could avoid the abstraction by choosing your own numbers and playing out the scenario that way. Imagine a  cubical box (volume ) being scaled down to a  cubical box (volume ). You'd go from a volume of  to a volume of , losing  of the volume, again equating to an  reduction.

The volume of the box is Base * Height, where Height is the length of the box and Base is the area of the triangular face. Thus, . The area of the base, then, is . Now recall that the area of an equilateral triangle is .

So . and  and , which is the value of .

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.