How to find the volume of a cube - High School Math

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Question

The density of gold is $16\ g/cm^3$ and the density of glass is $2\ g/cm^3$ . You have a gold cube that is $x\ cm$ in length on each side. If you want to make a glass cube that is the same weight as the gold cube, how long must each side of the glass cube be?

Answer

Weight = Density * Volume

Volume of Gold Cube = side3= x3

Weight of Gold = 16 g/cm3 * x3

Weight of Glass = 3/cm3 * side3

Set the weight of the gold equal to the weight of the glass and solve for the side length:

16* x3 = 2 * side3

side3 = 16/2* x3 = 8 x3

Take the cube root of both sides:

side = 2x

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Question

What is the sum of the number of vertices, edges, and faces of a cube?

Answer

Vertices = three planes coming together at a point = 8

Edges = two planes coming together to form a line = 12

Faces = one plane as the surface of the solid = 6

Vertices + Edges + Faces = 8 + 12 + 6 = 26

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Question

What is the difference in volume between a sphere with radius x and a cube with a side of 2x? Let π = 3.14

Answer

Vcube = s3 = (2x)3 = 8x3

Vsphere = 4/3 πr3 = 4/3•3.14•x3 = 4.18x3

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Question

A tank measuring 3in wide by 5in deep is 10in tall. If there are two cubes with 2in sides in the tank, how much water is needed to fill it?

Answer

$Volume_{Tank} = Width \cdot Depth \cdot Height$

$Volume_{Tank} = 3in \cdot 5in \cdot 10in = 150 in^3$

$Volume_{Cube} = (Side)^3 = (2in)^3 = 8 in^3$

$Volume_{Water}=Volume_{Tank}-2(Volume_{Cube})$

$Volume_{Water}=150-2(8)$

$Voume_{Water}=150-16$

$Volume_{Water}=134in^3$

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Question

If a cube has its edges increased by a factor of 5, what is the ratio of the new volume to the old volume?

Answer

A cubic volume is . Let the original sides be 1, so that the original volume is 1. Then find the volume if the sides measure 5. This new volume is 125. Therefore, the ratio of new volume to old volume is 125: 1.

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Question

A cube has edges that are three times as long as those of a smaller cube. The volume of the bigger cube is how many times larger than that of the smaller cube?

Answer

If we let represent the length of an edge on the smaller cube, its volume is $x^{3}$ .

The larger cube has edges three times as long, so the length can be represented as . The volume is $(3x)^{3}$ , which is $27x^{3}$ .

The large cube's volume of $27x^{3}$ is 27 times as large as the small cube's volume of $x^{3}$ .

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Question

What is the volume of a cube with a side length of 7?

Answer

When searching for the volume of a cube we are looking for the amount of the space enclosed by the cube.

To find this we must know the formula for the volume of a cube which is $Volume: of: Cube=s^{3}$

Using this formula we plug in the side length for to get $V=7^{3}$

Cube the side length to arrive at the answer of $7^{3}=343$

The answer is .

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Question

If the volume of a cube is 3375, what is the length of each side?

Answer

To find this we must know the formula for the volume of a cube which is

We plug the volume into the equation yielding $3375= s^{3}$

Then we take the cube root of both sides giving us $\sqrt[3]{3375}=\sqrt[3]{s^{3}}$

We now know

The answer is .

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Question

A box measures $11 \ cm \times 10 \ cm \times 8 \ cm$ . How many dice can fit in this box if the dice are cubes with sides of length $2 \ cm$ ?

Answer

Since the dice are $2 \ cm$ on each of their sides, of them would measure $10 \ cm$ in length when standing face-to-face. This means dice will fit along the edge of the box that measures $10 \ cm$ .

dice will also fit on the edge of the box measuring $11 \ cm$ , but will not fit since adding an additional die would bring the length of the dice standing face-to-face up to $12 \ cm$ . There are no half-dice to fill the gap, so there is a small empty space on this side.

dice will fit along the side that measures $8 \ cm$ .

Treating "dice" as the unit of measurement instead of and considering only the volume where the dice will fit, one ends up with a rectangular shape measuring dice $\times$ dice $\times$ dice. The volume of this shape is the number of dice that will fit in this area (and thus the box). Find the volume by multiplying all lengths of the shape's sides together:

$5 \times 5 \times 4$ dice dice.

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Question

What is the volume, in centimeters, of a rectangular prism with a length of , a width of , and a height of ?

Answer

In order to solve this problem, we need to make sure that the measurements given are in uniform units. In this case, we will use centimeters; therefore, we need to convert the other units of meters and millimeters into centimeters via dimensional analysis.

Solve for the width.

$w=\frac{255mm}{1}\frac{1cm}{10mm}$

Solve for the height.

$h=\frac{0.44m}{1}\frac{100cm}{1m}$

Now, solve for volume by using the formula .

$\rightarrow 2692.8cm^{3}$

If you calculated $\dpi{100}269.28 cm^{3}$ , then you did not convert into appropriate units.

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Question

What is the volume of a cube with a diagonal one one of its faces of $\dpi{100} d=6.27ft$ ?

Answer

A few facts need to be known to solve this problem. Observe that the diagonal of the square face of the cube cuts the qradilateral into two right isosceles triangles; therefore, the length of a side of the square to its diagonal is the same as an isosceles right triangle's leg to its hypotenuse: $1:\sqrt{2}$ .

$\frac{s}{d}=\frac{1}{\sqrt{2}}$

$\dpi{100} \dpi{100} \dpi{100} \frac{s}{6.27ft}=\frac{1}{\sqrt{2}}$

Rearrange an solve for .

$\dpi{100} \dpi{100} s=\frac{6.27ft}{\sqrt{2}}$

Now, solve for the volume.

$V=s^{3}$

$V=(\frac{6.27ft}{\sqrt{2}})(\frac{6.27ft}{\sqrt{2}})(\frac{6.27ft}{\sqrt{2}})$

$V=\frac{246.491883ft^{3}}{2\sqrt{2}}$

$\rightarrow87.15ft^{3}$

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Question

A building is in the shape of a perfect cube, with a side length of . What is the building's volume, in cubic meters?

Answer

In order to solve this question, we need to convert miles to meters via dimensional analysis and then solve for the volume of the building.

$s=\frac{0.44 mi}{1}\frac{1.60934km}{1mi}\frac{1000m}{1km}$

Now, solve for volume.

$V=s^{3}$

$V=(708.1096m)^{3}$

$\rightarrow 355059753.12m^{3}\rightarrow 355000000m^{3}$

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Question

This figure is a cube with one face having an area of 16 in2.

What is the volume of the cube (in3)?

Answer

The volume of a cube is one side cubed. Because we know that one face has an area of 16 in2, then we know that one side must be the square root of 16 or 4. Thus the volume is $4^{3}=64$ .

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Question

A cube has a surface area of units squared. What is its volume?

Answer

Since a cube has square faces and the surface area of each face is given by multiplying the length of one side of the square face by itself, the equation for the surface area of a cube is , where is the length of one side of one face (i.e., one edge of the cube). Find the length of one side/edge of the given cube by setting the given surface area equal to :

$\rightarrow s^2 = 49$

$\rightarrow s = 7$ units

The volume of a cube is , where is the length of one of the cube's edges. Substituing the solution to the previous equation for in the volume equation gives the volume of the cube:

$\rightarrow V = (7)^3$

$\rightarrow V = 343$ units cubed

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Question

A cube has a height of 4 feet. What is the volume of the cube in feet?

Answer

We only have the length of one edge, but that is sufficient for finding the volume. The volume of a cube equals one edge multiplied by itself three times: $e^{3}$

$4^{3}=64$

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Question

Find the volume of the following cube:

Answer

The formula for the volume of a cube is

,

where is the side of the cube.

Plugging in our values, we get:

$V=(4\ m)^3$

$V=64\ m^3$

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Question

A cube has a side length of meters. What is the volume of the cube?

Answer

The formula for the volume of a cube is:

$Volume = (side)^{3}$

Since the length of one side is meters, the volume of the cube is:

$2^{3 } = 8$ meters cubed.

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Question

If the length of the side of a cube is $8x^{2}$ , which expression represents the volume of the cube?

Answer

The formula for the volume of the cube is $V = x^{3}$

Plugging that into Volume equation, we find $8^{3}$ and $(x^{^{2)^{3}}}$

Thus, the answer is 512x6

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