Cubes - ISEE Upper Level Quantitative Reasoning

One of the faces of the cube could be drawn like this:

Notice that this makes a triangle.

This means that we can create a proportion for the sides. On the standard triangle, the non-hypotenuse sides are both , and the hypotenuse is . This will allow us to make the proportion:

Multiplying both sides by , you get:

To find the area of the square, you need to square this value:

Now, since there are sides to the cube, multiply this by to get the total surface area:

One and one half feet is equal to eighteen inches, so the volume of the cube, in cubic inches, is the cube of this, or

cubic inches.

The volume of a rectangular prism is

Since its volume is the same as that of the cube, and its length and height are 27 and 9 inches, respectively, we can rewrite this as

The width is 24 inches.

One and one half feet is equal to eighteen inches, so the surface area of the cube, in square inches, is six times the square of this, or

square inches.

The surface area of a rectangular prism is determined by the formula

.

So, with substitutiton, we can find the width:

inches

The volume of a rectangular prism is the product of its length, its width, and its height. The prism's volume of one cubic foot is equal to cubic inches.

Therefore, can be rewritten as .

We can solve for as follows:

The cube with the greater sidelength has the greater volume, so we need only calculate and compare sidelengths.

(a) , so the sidelength of the first cube can be found as follows:

inches

(b) by an extension of the Pythagorean Theorem, so the sidelength of the second cube can be found as follows:

Since , . The second cube has the greater sidelength and, subsequently, the greater volume. This makes (b) greater.

The volume of a pyramid or a cone with height and base of area is

,

so in both cases, the area of the base is

Since the pyramid and the cone have the same volume and height, their bases has the same area .

The length of one side of the square base of the pyramid is the square root of this, or , and the perimeter is four times this, or .

The radius and the area of the base of the cone are related as follows:

Multiply both sides by to get:

, so

, and

The perimeter of the base of the pyramid, which is (A), is greater than the circumference of the base of the cone.

The sidelengths of Cubes 1, 2, 3, and 4 can be given values , respectively.

Then the volumes of the cubes are as follows:

Cube 1:

Cube 2:

Cube 3:

Cube 4:

In both answer choices ask for a mean, so we can determine which answer (mean) is greater simply by comparing the sums of volumes.

(a) The sum of the volumes of Cubes 1 and 4 is .

(b) The sum of the volumes of Cubes 2 and 3 is .

Regardless of , the sum of the volumes of Cubes 1 and 4 is greater, and therefore, so is their mean.

It is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:

(Note that not all points are drawn in on the cube.)

The two points we are looking at are:

and

Solve this by using the distance formula. This is very easy since one point is all s. It is merely:

This is approximately .

It is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:

(Note that not all points are drawn in on the cube.)

The two points we are looking at are:

and

Solve this by using the distance formula. This is very easy since one point is all s. It is merely:

This is approximately .

First, you need to find the side length of this cube. We know that the volume is:

, where is the side length.

Therefore, based on our data, we can say:

Solving for by taking the cube-root of both sides, we get:

Now, it is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:

(Note that not all points are drawn in on the cube.)

The two points we are looking at are:

and

Solve this by using the distance formula. This is very easy since one point is all s. It is merely:

This is approximately .

First, you need to find the side length of this cube. We know that the surface area is defined by:

, where is the side length. (This is because the cube is sides of equal area).

Therefore, based on our data, we can say:

Take the square root of both sides and get:

Now, it is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:

(Note that not all points are drawn in on the cube.)

The two points we are looking at are:

and

Solve this by using the distance formula. This is very easy since one point is all s. It is merely:

This is approximately .

One of the faces of the cube could be drawn like this:

Notice that this makes a triangle.

This means that we can create a proportion for the sides. On the standard triangle, the non-hypotenuse sides are both , and the hypotenuse is . This will allow us to make the proportion:

Multiplying both sides by , you get:

Recall that the formula for the volume of a cube is:

Therefore, we can compute the volume using the side found above:

Now, rationalize the denominator:

We must begin by using the equation for the volume of a cube:

(It is like doing the area of a square, then adding another dimension!)

We know that the volume is . Therefore, we can rewrite our equation:

Using your calculator, we can find the cube root of . It is . (If you get just round up to . This is a calculator issue!).

This is the side length you need!

Another way you could do this is by cubing each of the possible answers to see which gives you a volume of .

Recall that the formula for the surface area of a cube is:

, where is the length of a side of the cube. This equation is easy to memorize because it is merely a multiplication of a single side () by because a cube has equal sides.

Now, we know that is ; therefore, we can write:

Solve for :

Take the square root of both sides:

This is the length of one of your sides.

The volume of a cube is defined by the formula

where is the length of one side.

If , then

and

So one side measures 7 inches.

The surface area of a cube is defined by the formula

, so

The surface area is 294 square inches.

We know that the volume of a cube can be found with the equation:

, where is the side length of the cube.

Now, if the volume is , then we know:

Either with your calculator or with careful math, you can solve by taking the cube-root of both sides. This gives you:

This means that each side of the cube is long; therefore, each face has an area of , or . Since there are sides to a cube, this means the total surface area is , or .

This question is very easy. Do not over-think it! All you need to do is calculate the area of one side of the cube. Then, multiply that by (since the cube has sides). Each side of a cube is, of course, a square; therefore, the area of one side of this cube is , or . This means that the whole cube has a surface area of or .

Recall that the formula for the surface area of a cube is:

, where is the length of a side of the cube. This equation is easy to memorize because it is merely a multiplication of a single side () by because a cube has equal sides.

For our data, we know that ; therefore, our equation is:

To solve this, first calculate the side length based on the volume given. Recall that the equation for the volume of a cube is:

, where is the side length.

For our data, this gives us:

Taking the cube-root of both sides, we get:

Now, use the surface area formula to compute the total surface area:

, where is the length of a side of the cube. This equation is easy to memorize because it is merely a multiplication of a single side () by because a cube has equal sides.

For our data, this gives us:

To solve this, first calculate the side length based on the volume given. Recall that the equation for the volume of a cube is:

, where is the side length.

For our data, this gives us:

Taking the cube-root of both sides, we get:

(You will need to use a calculator for this. If your calculator gives you something like . . . it is okay to round. This is just the nature of taking roots!).

Now, use the surface area formula to compute the total surface area:

, where is the length of a side of the cube. This equation is easy to memorize because it is merely a multiplication of a single side () by because a cube has equal sides.

For our data, this gives us: