Problem 66 - Calculus: One and Several Variables

Question

Express the volume of a cube as a function of the total surface area.

Step-by-Step Solution

Answer

The volume of a cube can be expressed as a function of the total surface area as \(V = (\sqrt{A/6})^3\).

Step 1: Remember the formula for the surface area of a cube.

The total surface area \(A\) of a cube is given by \(A = 6a^2\), where \(a\) is the side length of the cube.

Step 2: Express the length of the side as a function of the surface area.

From the formula \(A = 6a^2\), we can express the side length as a function of the surface area. This gives us \(a = \sqrt{A/6}\).

Step 3: Substitute the expression of \(a\) in the volume formula.

The volume of a cube is given by \(V = a^3\). Substituting the expression of \(a\) derived previously, we have \(V = (\sqrt{A/6})^3\).

Key Concepts

Volume of a Cube

A cube is a three-dimensional shape with six equal square faces. Its volume is a measure of the capacity within the cube and is computed using the formula:

• Volume (\(V\)) = \(a^3\)

where \(a\) is the length of any of the cube's sides.
The volume helps us understand how much space is enclosed within the cube. For example, if each side of a cube is 3 units long, its volume is \(3^3 = 27\) cubic units. This means it can hold 27 unit cubes inside.Knowing how to express volume using different given characteristics like surface area adds another layer of understanding. This expression can be particularly useful in problems requiring optimization or comparison of different geometric properties.

Surface Area

Surface area refers to the total area covered by the faces of the cube. Since a cube consists of six identical square faces, its total surface area is expressed as:

• Surface Area (\(A\)) = \(6a^2\)

where \(a\) is the length of one side of the cube.
By knowing the surface area, you can determine how much material is needed to cover the cube without gaps. Now, if you know the surface area but not the side length, you can rearrange the formula and solve for \(a\). This involves dividing the total surface area by 6, then taking the square root:

• \(a = \sqrt{\frac{A}{6}}\)

Understanding this relationship allows solving different problems, such as designing packaging and optimizing the use of materials.

Function of a Variable

A function describes a relationship between two quantities, where one quantity depends on the other. When the volume of a cube is expressed as a function of its surface area, it allows you to see how changes in surface area affect the volume.Consider the equation for volume when expressed in terms of surface area:

• Volume \(V\) = \((\sqrt{\frac{A}{6}})^3\)

Here, \(A\) is treated as a variable. The function tells us that if the surface area \(A\) changes, the volume \(V\) changes too.
This is particularly useful in calculus, where functions help analyze and interpret rates of change and optimize different properties. By understanding how these variables interact, one can predict and control outcomes in various engineering, architectural, and physical scenarios.