Cubes - GRE Quantitative Reasoning
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The surface area of a cube is 486 units. What is the distance of its diagonal (e.g. from its front-left-bottom corner to its rear-right-top corner)?
The surface area of a cube is 486 units. What is the distance of its diagonal (e.g. from its front-left-bottom corner to its rear-right-top corner)?
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)
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)
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You have a rectangular box with dimensions 6 inches by 6 inches by 8 inches. What is the length of the shortest distance between two non-adjacent corners of the box?
You have a rectangular box with dimensions 6 inches by 6 inches by 8 inches. What is the length of the shortest distance between two non-adjacent corners of the box?
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 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
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What is the length of the diagonal of a cube with side lengths of 
each?
What is the length of the diagonal of a cube with side lengths of
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 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:
Now, if the the value of
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What is the length of the diagonal of a cube that has a surface area of 
?
What is the length of the diagonal of a cube that has a surface area of
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 
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
Therefore, for our data, we have:
Solving for
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:
Now, if the the value of
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Quantity A: The length of a side of a cube with a volume of 
.
Quantity B: The length of a side of a cube with surface area of 
.
Which of the following is true?
Quantity A: The length of a side of a cube with a volume of
Quantity B: The length of a side of a cube with surface area of
Which of the following is true?
Recall that the equation for the volume of a cube is:
Since the sides of a cube are merely squares, the surface area equation is just 
times the area of one of those squares:
So, for our two quantities:
Quantity A
Use your calculator to estimate this value (since you will not have a square root key). This is 
.
Quantity B
First divide by 
:
Therefore, 
Therefore, the two quantities are equal.
Recall that the equation for the volume of a cube is:
Since the sides of a cube are merely squares, the surface area equation is just
So, for our two quantities:
Quantity A
Use your calculator to estimate this value (since you will not have a square root key). This is
Quantity B
First divide by
Therefore,
Therefore, the two quantities are equal.
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What is the length of an edge of a cube with a surface area of 
?
What is the length of an edge of a cube with a surface area of
The surface area of a cube is made up of 
squares. Therefore, the equation is merely 
times the area of one of those squares. Since the sides of a square are equal, this is:
, where 
is the length of one side of the square.
For our data, we know:
This means that:
Now, while you will not have a calculator with a square root key, you do know that 
. (You can always use your calculator to test values like this.) Therefore, we know that 
. This is the length of one side
The surface area of a cube is made up of
For our data, we know:
This means that:
Now, while you will not have a calculator with a square root key, you do know that
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If a cube has a total surface area of 
square inches, what is the length of one edge?
If a cube has a total surface area of
There are 6 sides to a cube. If the total surface area is 54 square inches, then each face must have an area of 9 square inches.
Every face of a cube is a square, so if the area is 9 square inches, each edge must be 3 inches.
There are 6 sides to a cube. If the total surface area is 54 square inches, then each face must have an area of 9 square inches.
Every face of a cube is a square, so if the area is 9 square inches, each edge must be 3 inches.
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A rectangular box is 6 feet wide, 3 feet long, and 2 feet high. What is the surface area of this box?
A rectangular box is 6 feet wide, 3 feet long, and 2 feet high. What is the surface area of this box?
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 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.
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What is the surface area of a rectangular box that is 3 feet high, 6 feet long, and 4 feet wide?
What is the surface area of a rectangular box that is 3 feet high, 6 feet long, and 4 feet wide?
Surface area of a rectangular solid
= 2_lw_ + 2_lh_ + 2_wh_
= 2(6)(4) + 2(6)(3) + 2(4)(3)
= 108
Surface area of a rectangular solid
= 2_lw_ + 2_lh_ + 2_wh_
= 2(6)(4) + 2(6)(3) + 2(4)(3)
= 108
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A large cube is made by fitting 8 smaller, identical cubes together. If the volume of each of the smaller cubes is 27, what is the surface area of the large cube?
A large cube is made by fitting 8 smaller, identical cubes together. If the volume of each of the smaller cubes is 27, what is the surface area of the large cube?
Since the volume of the smaller cubes with edges, 
, is 27, we have:
.
The large cube has edges 
.
So the surface area of the large cube is:
.
Since the volume of the smaller cubes with edges,
The large cube has edges
So the surface area of the large cube is:
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Quantity A:
The surface area of a cube with a volume of 
.
Quantity B:
The volume of a cube with a surface area of 
.
Quantity A:
The surface area of a cube with a volume of
Quantity B:
The volume of a cube with a surface area of
The relationship can be determined, because it is possible to find the surface area of a cube from the volume and vice versa.
Quantity A:
To find the surface area of the cube, you must find the side length. To find the side length from the volume, you must find the cube root.
Find the cube root of the volume.
Insert into surface area equation.
Quantity B:
To find the volume of a cube, you must find the side length. To find the side length from the surface area, you must divide by 6, then find the square root of the result. Then, cube that result.
Divide by 6.
Square root.
Now, to find the volume.
Quantity B is greater.
The relationship can be determined, because it is possible to find the surface area of a cube from the volume and vice versa.
Quantity A:
To find the surface area of the cube, you must find the side length. To find the side length from the volume, you must find the cube root.
Find the cube root of the volume.
Insert into surface area equation.
Quantity B:
To find the volume of a cube, you must find the side length. To find the side length from the surface area, you must divide by 6, then find the square root of the result. Then, cube that result.
Divide by 6.
Square root.
Now, to find the volume.
Quantity B is greater.
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What is the volume of a rectangular box that is twice as long as it is high, and four times as wide as it is long?
What is the volume of a rectangular box that is twice as long as it is high, and four times as wide as it is long?
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.
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.
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What is the volume of a cube with a surface area of 
?
What is the volume of a cube with a surface area of
The surface area of a cube is merely the sum of the surface areas of the 
squares that make up its faces. Therefore, the surface area equation understandably is:
, where 
is the side length of any one side of the cube. For our values, we know:
Solving for 
, we get:
or 
Now, the volume of a cube is defined by the simple equation:
For 
, this is:
The surface area of a cube is merely the sum of the surface areas of the
Solving for
Now, the volume of a cube is defined by the simple equation:
For
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The volume of a cube is 
. If the side length of this cube is tripled, what is the new volume?
The volume of a cube is
Recall that the volume of a cube is defined by the equation:
, where 
is the side length of the cube.
Therefore, if we know that 
, we can solve:
This means that 
.
Now, if we triple 
to 
, the new volume of our cube will be:
Recall that the volume of a cube is defined by the equation:
Therefore, if we know that
This means that
Now, if we triple
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What is the volume of a cube with surface area of 
?
What is the volume of a cube with surface area of
Recall that the equation for the surface area of a cube is merely derived from the fact that the cube's faces are made up of 
squares. It is therefore:
For our values, this is:
Solving for 
, we get:
, so 
Now, the volume of a cube is merely:
Therefore, for 
, this value is:
Recall that the equation for the surface area of a cube is merely derived from the fact that the cube's faces are made up of
For our values, this is:
Solving for
Now, the volume of a cube is merely:
Therefore, for
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A cube has a volume of 64, what would it be if you doubled its side lengths?
A cube has a volume of 64, what would it be if you doubled its side lengths?
To find the volume of a cube, you multiple your side length 3 times (s*s*s).
To find the side length from the volume, you find the cube root which gives you 4
.
Doubling the side gives you 8
.
The volume of the new cube would then be 512
.
To find the volume of a cube, you multiple your side length 3 times (s*s*s).
To find the side length from the volume, you find the cube root which gives you 4
Doubling the side gives you 8
The volume of the new cube would then be 512
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