Solid Geometry - ISEE Upper Level Quantitative Reasoning

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

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 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 .

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:

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:

Now, this could look like a difficult problem; however, think of the equation for finding the length of the diagonal of a cube. It is like the Pythagorean Theorem, just adding an additional dimension:

(It is very easy, because the three lengths are all the same: ).

So, we know this, then:

To solve, you can factor out an from the root on the right side of the equation:

Just by looking at this, you can tell that the answer is:

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 is:

Now, this could look like a difficult problem. However, think of the equation for finding the length of the diagonal of a cube. It is like the Pythagorean Theorem, just adding an additional dimension:

(It is very easy, because the three lengths are all the same: ).

So, we know this, then:

To solve, you can factor out an from the root on the right side of the equation:

Just by looking at this, you can tell that the answer is:

Now, use the equation for the volume of a cube:

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

For our data, it is:

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 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.

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:

This question is relatively straightforward. The equation for the volume of a cube is:

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

Now, for our data, we merely need to "plug and chug:"