Solid Geometry - GRE Quantitative Reasoning

circumference = πd

d = 2r

volume of cylinder = πr2h

r = 8, h = 4

volume = 256π

Begin by solving for the length of one side of the cube. Use the formula for surface area to do this:

s= length of one side of the cube

The length of the side of the cube is equal to the diameter of the sphere. Therefore, the radius of the sphere is 3. Now use the formula for the surface area of a sphere:

The surface area of the sphere is .

The surface area formula we need to solve this is 2_ab_ + 2_bc_ + 2_ac_. So if we let a = 6, b = 3, and c = 2, then surface area:

= 2(6)(3) + 2(3)(2) +2(6)(2)

= 72 sq ft.

The box is 2 times as long as it is high, so H = L/2. It is also 4 times as wide as it is long, so W = 4_L_. Now we need volume = L * W * H = L * 4_L_ * L/2 = 2_L_3.

Begin by solving for the length of one side of the cube. Use the formula for surface area to do this:

s= length of one side of the cube

The length of the side of the cube is equal to the diameter of the sphere. Therefore, the radius of the sphere is 3. Now use the formula for the surface area of a sphere:

The surface area of the sphere is .

Surface Area = 4_πr_2 = 4 * 22/7 * 72 = 616

Volume of a sphere = 4/3 * πr_3 = 4/3 * π * 33 = 36_π

The shortest length between any two non-adjacent corners will be the diagonal of the smallest face of the rectangular box. The smallest face of the rectangular box is a six-inch by six-inch square. The diagonal of a six-inch square is .

The diagonal length of a cube is found by a form of the distance formula that is akin to the Pythagorean Theorem, though with an additional dimension added to it. It is:

, or , or

Now, if the the value of is , we get simply

To begin, the best thing to do is to find the length of a side of the cube. This is done using the formula for the surface area of a cube. Recall that a cube is made up of squares. Therefore, its surface area is:

, where is the length of a side.

Therefore, for our data, we have:

Solving for , we get:

This means that

Now, the diagonal length of a cube is found by a form of the distance formula that is akin to the Pythagorean Theorem, though with an additional dimension added to it. It is:

, or , or

Now, if the the value of is , we get simply

First, we must ascertain the length of each side. Based on our initial data, we know that the 6 faces of the cube will have a surface area of 6x2. This yields the equation:

6x2 = 486, which simplifies to: x2 = 81; x = 9.

Therefore, each side has a length of 9. Imagine the cube is centered on the origin. This means its "front-left-bottom corner" will be at (–4.5, –4.5, 4.5) and its "rear-right-top corner" will be at (4.5, 4.5, –4.5). To find the distance between these, we use the three-dimensional distance formula:

d = √((x1 – x2)2 + (y1 – y2)2 + (z1 – z2)2)

For our data, this will be:

√( (–4.5 – 4.5)2 + (–4.5 – 4.5)2 + (4.5 + 4.5)2) =

√( (–9)2 + (–9)2 + (9)2) = √(81 + 81 + 81) = √(243) =

√(3 * 81) = √(3) * √(81) = 9√(3)

The shortest length between any two non-adjacent corners will be the diagonal of the smallest face of the rectangular box. The smallest face of the rectangular box is a six-inch by six-inch square. The diagonal of a six-inch square is .

The diagonal length of a cube is found by a form of the distance formula that is akin to the Pythagorean Theorem, though with an additional dimension added to it. It is:

, or , or

Now, if the the value of is , we get simply

To begin, the best thing to do is to find the length of a side of the cube. This is done using the formula for the surface area of a cube. Recall that a cube is made up of squares. Therefore, its surface area is:

, where is the length of a side.

Therefore, for our data, we have:

Solving for , we get:

This means that

Now, the diagonal length of a cube is found by a form of the distance formula that is akin to the Pythagorean Theorem, though with an additional dimension added to it. It is:

, or , or

Now, if the the value of is , we get simply

First, we must ascertain the length of each side. Based on our initial data, we know that the 6 faces of the cube will have a surface area of 6x2. This yields the equation:

6x2 = 486, which simplifies to: x2 = 81; x = 9.

Therefore, each side has a length of 9. Imagine the cube is centered on the origin. This means its "front-left-bottom corner" will be at (–4.5, –4.5, 4.5) and its "rear-right-top corner" will be at (4.5, 4.5, –4.5). To find the distance between these, we use the three-dimensional distance formula:

d = √((x1 – x2)2 + (y1 – y2)2 + (z1 – z2)2)

For our data, this will be:

√( (–4.5 – 4.5)2 + (–4.5 – 4.5)2 + (4.5 + 4.5)2) =

√( (–9)2 + (–9)2 + (9)2) = √(81 + 81 + 81) = √(243) =

√(3 * 81) = √(3) * √(81) = 9√(3)

The shortest length between any two non-adjacent corners will be the diagonal of the smallest face of the rectangular box. The smallest face of the rectangular box is a six-inch by six-inch square. The diagonal of a six-inch square is .

The diagonal length of a cube is found by a form of the distance formula that is akin to the Pythagorean Theorem, though with an additional dimension added to it. It is:

, or , or

Now, if the the value of is , we get simply

To begin, the best thing to do is to find the length of a side of the cube. This is done using the formula for the surface area of a cube. Recall that a cube is made up of squares. Therefore, its surface area is:

, where is the length of a side.

Therefore, for our data, we have:

Solving for , we get:

This means that

Now, the diagonal length of a cube is found by a form of the distance formula that is akin to the Pythagorean Theorem, though with an additional dimension added to it. It is:

, or , or

Now, if the the value of is , we get simply

The surface area of a sphere is defined by the equation:

For our data, this means:

Solving for , we get:

or

The diameter of the sphere is .

Do not assume that the diameter will be half of the diameter of a sphere with volume of . Instead, begin with the sphere with a volume of . Such a simple action will prevent a vexing error!

Thus, we know from our equation for the volume of a sphere that:

Solving for , we get:

If you take the cube-root of both sides, you have:

First, you can factor out an :

Next, factor the :

Which simplifies to:

Thus, the diameter is double that or: