Solid Geometry - Math

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Question

Find the length of an edge of the following cube:

The volume of the cube is $27\ m^3$ .

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Answer

The formula for the volume of a cube is

,

where is the length of the edge of a cube.

Plugging in our values, we get:

$27\ m^3 = (e)^3$

$e = 3\ m$

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Question

Find the length of an edge of the following cube:

The volume of the cube is $64\ m^3$ .

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Answer

The formula for the volume of a cube is

,

where is the length of the edge of a cube.

Plugging in our values, we get:

$64\ m^3=(e)^3$

$e=4\ m$

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Question

What is the length of an edge of a cube that has a surface area of 54?

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Answer

The surface area of a cube can be determined using the following equation:

$SA=6\times l^2$

$\frac{54}{6}=\frac{6l^2}{6}$

$\sqrt9=\sqrt{l^2}$

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Question

If the surface area of a cube equals 96, what is the length of one side of the cube?

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Answer

The surface area of a cube = 6a2 where a is the length of the side of each edge of the cube. Put another way, since all sides of a cube are equal, a is just the lenght of one side of a cube.

We have 96 = 6a2 → a2 = 16, so that's the area of one face of the cube.

Solving we get √16, so a = 4

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Question

A sphere with a volume of $\frac{32}{3}$ $\pi m^{3}$ is inscribed in a cube, as shown in the diagram below.

What is the surface area of the cube, in $m^{2}$ ?

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Answer

We must first find the radius of the sphere in order to solve this problem. Since we already know the volume, we will use the volume formula to do this.

$V_{sphere}=\frac{4}{3}\pi r^{3}$

$\frac{32}{3}\pi =\frac{4}{3}\pi r^{3}$

$\frac{32}{3}=\frac{4}{3}r^{3}$

$\frac{3}{4}\cdot \frac{32}{3}= r^{3}$

$8=r^{3}$

With the radius of the sphere in hand, we can now apply it to the cube. The radius of the sphere is half the distance from the top to the bottom of the cube (or half the distance from one side to another). Therefore, the radius represents half of a side length of a square. So in this case

$side=2\cdot 2=4$

The formula for the surface area of a cube is:

$SA_{cube}=6s^{2}$

$6s^{2}=6\cdot (4)^{2}=6\cdot 16=96$

The surface area of the cube is $96\ m^{2}$

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Question

The side of a cube has a length of $\dpi{100} \small 5 cm$ . What is the total surface area of the cube?

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Answer

A cube has 6 faces. The area of each face is found by squaring the length of the side.

$\dpi{100} \small 5\times 5 = 25$

Multiply the area of one face by the number of faces to get the total surface area of the cube.

$\dpi{100} \small 25 \times 6=150$

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Question

What is the surface area of a cube with a side length of ?

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Answer

To find the surface area of a cube, we must count the number of surface faces and add the areas of each together. In a cube there are faces, each a square with the same side lengths. In this example the side length is .

The area of a square is given by the equation . Using our side length, we can solve the area of once face of the cube.

$A=(7)^{2}=49$

We then multiply this number by , the number of faces of the cube to find the total surface area.

Our answer for the surface area is .

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Question

What is the surface area of a cube with a side length of 15?

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Answer

To find the surface area of a cube we must count the number of surface faces and add the areas of each of them together.

In a cube there are 6 faces, each a square with the same side lengths.

In this example the side lengths is 15 so the area of each square would be

We then multiply this number by 6, the number of faces of the cube, to get

Our answer for the surface area is .

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Question

If a right triangle has a hypotenuse of length 5, and the length of the other sides are and , what would be the surface area of a cube having side length ?

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Answer

By the Pythagorean Theorem,

$x=\sqrt{5}$

The surface area of a cube having 6 sides, is 6 times the area of one of its sides.

The area of any side of a cube is the square of the side length.

So if the side length is , the area of any side is , or .

Thus the surface area of the cube is

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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?

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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

What is the surface area of a cube with a side length of ?

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Answer

In order to find the surface area of a cube, use the formula $SA=6(s^{2})$ .

$SA=6(7.2in)^{2}$

$SA=651.84in^{2}$

$SA=311.04in^{2}$

$\rightarrow 311in^{2}$

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Question

What is the surface area, in inches, of a rectangular prism with a length of $\dpi{100} l=2.2ft$ , a width of , and a height of ?

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Answer

In order to find the surface area of a rectangular prism, use the formula .

However, all units must be the same. All of the units of this problem are in inches with the exception of $\dpi{100} l=2.2ft$ .

Convert to inches.

$l=\frac{2.2ft}{1}*\frac{12in}{1ft}$

Now, we can insert the known values into the surface area formula in order to calulate the surface area of the rectangular prism.

$SA=(950.4in^{2})+(504in^{2})+(739.2in^{2})$

$SA=2193.6in^{2}$

If you calculated the surface area to equal $6652.8in^{2}$ , then you utilized the volume formula of a rectangular prism, which is ; this is incorrect.

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Question

$\textup{What is the surface area, in terms of }p,\textup{of a cube with sides of length }p\textup{ ?}$

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Answer

$\textup{Each of the six identical faces of the cube is a square with sides }p.$

$\textup{Each face: }A=p^{2}: : : : : \textup{Surface area}=6p^{2}$

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Question

What is the surface area of a cube with a diagonal of $\dpi{100} d=4.2cm$ ?

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Answer

A few facts need to be known to solve this problem. Observe that the diagonal of the square face of the cube cuts it 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: $1:\sqrt{2}$ .

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

$\dpi{100} \dpi{100} \frac{s}{4.2cm}=\frac{1}{\sqrt{2}}$

Rearrange an solve for .

$\dpi{100} s=\frac{4.2cm}{\sqrt{2}}$

Now, solve for the area of the cube using the formula $\dpi{100} \dpi{100} SA=6(s^{2})$ .

$SA=6(\frac{4.2cm}{\sqrt{2}})^{2}$

$SA=\frac{6}{1}(\frac{4.2cm}{\sqrt{2}}\frac{4.2cm}{\sqrt{2}})$

$\dpi{100} SA=\frac{6}{1}(\frac{17.64cm^{2}}{2})$

$\dpi{100} \dpi{100} SA=3*17.64cm^{2}$

$\rightarrow 52.92cm^{2}$

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Question

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

What is the surface area of the cube (in2)?

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Answer

The surface area of a cube is the sum of the area of each face. Since there are 6 faces on a cube, the surface area of the entire cube is $6\cdot16$ .

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Question

The side length of a particular cube is $\frac{3}{2}$ . What is the surface area of this cube?

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Answer

To find the surface of a cube, use the standard equation:

where denotes the side length.

Plug in the given value for to find the answer:

$SA=6\cdot \left(\frac{3}{2}\right)^2=6\cdot \left(\frac{9}{4}\right)=\frac{27}{2}$

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Question

Sarah is wrapping a birthday present. The box is a cube with sides of . At a minimum, how many square feet of wrapping paper will she need?

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Answer

Remember, .

For a cube:

$SA = 6s^{2}$

Thus $6(0.50)^{2}=6(0.25)=1.50; ft^{2}$ .

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Question

Find the surface area of the following cube:

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Answer

The formula for the surface area of a cube is

,

where is the length of the side.

Plugging in our values, we get:

$SA=6(4\ m)^2$

$SA=96\ m^2$

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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?

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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?

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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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