Using the volume given, we take it's cube to find the length of the cube: $\displaystyle ^3\sqrt{512}=8$ cm.

Therefore, the length of the cube is 8 cm.

Knowing the properties of a cube, this implies that the width and height of the cube is also 8 cm.

Since all sides are identical, the formula for the surface area is length times width times the number of sides: $\displaystyle 8\cdot8\cdot6=384 cm^2$.

Write the formula to find the surface of a cube, where $\displaystyle x$ is the length.

$\displaystyle A=6x^2$

Substitute and solve.

$\displaystyle A=6(1\:cm)^2=6\:cm^2$

Write the formula for the surface area of a cube, substitute the length provided in the question, and simplify.

$\displaystyle A=6x^2 = 6(\sqrt2\:in)^2 = 6\cdot2\:in^2 =12\:in^2$

The formula for the surface area of a cube is:

$\displaystyle SA=6s^2$, where $\displaystyle s$ is the length of one side of the cube.

We are given the length of one side of the cube in question, so we can substitute that value into the surface area equation and solve:

$\displaystyle SA=6s^2 = 6(x+3)^2 = 6(x^2+6x+9) =6x^2+36x+54$

Since the sphere is inscribed within the cube, its diameter is the same length as an edge of the cube. Since cubes have identical side lengths we find the area of one side and then multiply by the number of sides to find the total surface area.

Area of one side:

$\displaystyle A=s*s=4*4=16\hspace{1mm}m^2$

Total surface area:

$\displaystyle SA=16\hspace{1mm}m^2*6=\mathbf{96\hspace{1mmm^2}}$

We first need to know the edge length before we can solve for surface area. Since we are provided the volume and all edges are of equal length, we can use the formula for volume to get the length of sides:

$\displaystyle \\volume=length\cdot width\cdot height\\ \\27\;cm^3=a^3\\ \\a=\sqrt[3]{27\;cm^3}\\ \\a=3\;cm$

Now that we know the length of sides, we can plug this value into our surface area formula:

$\displaystyle \\S.A.=6a^2\\S.A.=6(3\;cm)^2\\S.A.=6\cdot9\;cm^2=54\;cm^2$

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

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

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

The volume of a cube is

$\displaystyle V = s^3$

So with a side length of 7 inches, the volume is

$\displaystyle V = 7^{3}\, in^3 = 343\, in^3$

The volume of a cube is

$\displaystyle V = s^3$

So with a side length of 2 ft, the volume is

$\displaystyle V = 2^{3}\, ft^3 = 8\, ft^3$

The volume of a cube can be determined through the equation $\displaystyle V = s^3$, where $\displaystyle s$ stands for the length of one side of the cube. The equation is $\displaystyle s\cdot s\cdot s$ because all edges in a cube are the same length. The value for the given edge just needs to be substituted into the equation for $\displaystyle s$ in order to solve for the cube's volume.

$\displaystyle V = s^3$

$\displaystyle V = 7^3$

$\displaystyle V = 7 \cdot 7\cdot 7$

$\displaystyle V = 343\:in^3$