Solid Geometry - Math

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:

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 surface area of a rectangular prism can be determined using the formula:

Using substitutions, the surface areas of the prisms can be found as follows:

The prism in (A):

Regardless of the value of , - that is, the first prism has the greater surface area. (A) is greater.

Divide each dimension by 3 to convert feet to yards, then multiply the three dimensions together:

The volume of a rectangular solid ten inches by twenty inches by fifteen inches is

cubic inches.

The volume of a cube with sidelength 13 inches is

cubic inches.

This makes (B) greater

Since the answer is not dependent on the actual dimensions, for the sake of simplicity, we assume that the base has sidelength 2. Then the area of the base is the square of this, or 4.

The height of the pyramid is equal to the perimeter of the base, or . A right triangle can be formed with the lengths of its legs equal to the height of the pyramid, or 8, and one half the length of a side, or 1; the length of its hypotenuse, which is the slant height, is

This is the height of one triangular face; its base is a side of the square, so the length of the base is 2. The area of a face is half the product of these dimensions, or

Since twice the area of the base is , the problem comes down to comparing and ; the latter, which is (B), is greater.

Use the formula on each pyramid.

(a)

(b)

Regardless of , (b) is the greater quantity.

The volume of a pyramid with height and a square base with sidelength is

.

(a) Substitute :

(b) Substitute :

The two pyramids have equal volume.

The volume of a pyramid is one-third of the product of the height and the area of the base. The areas of the bases can be calculated, but no information is given about the heights of the pyramids. There is not enough information to determine which one has the greater volume.

The volume of a pyramid is given by the formula

where is the area of its base and is its height.

Let be the length of one side of the square base. Then the height is equal to the perimeter of that square, so

and the area of the base is

So the volume formula becomes

Solve for :

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:

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: