How can we express the surface area of a cube in terms of its volume?

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Understanding the Relationship Between Surface Area and Volume of a Cube

A cube is a three-dimensional shape where all sides are equal in length. To express the surface area of a cube as a function of its volume, we first need to recall the formulas for both the surface area and volume of a cube.

Formulas for Cube’s Volume and Surface Area

  • Volume (V): The volume of a cube is calculated using the formula V = a^3, where a is the length of one side of the cube.
  • Surface Area (A): The surface area of a cube is given by the formula A = 6a^2.

Deriving Surface Area as a Function of Volume

To express the surface area in terms of the volume, we need to manipulate these equations.

  1. Start with the volume formula: V = a^3.
  2. To find a, take the cube root of the volume: a = V^{1/3}.
  3. Now substitute a into the surface area formula:
  4. A = 6a^2 = 6(V^{1/3})^2
  5. Which simplifies to: A = 6V^{2/3}.

Final Expression

Thus, the surface area of a cube can be expressed as a function of its volume:

A(V) = 6V^{2/3}

This shows that as the volume of the cube changes, we can calculate the corresponding surface area using this formula.

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