Problem 13 - Thomas Calculus

Question

Express the edge length of a cube as a function of the cube's diagonal length \(d .\) Then express the surface area and volume of the cube as a function of the diagonal length.

Step-by-Step Solution

Answer

Edge length: \(\frac{d}{\sqrt{3}}\), Surface area: \(2d^2\), Volume: \(\frac{d^3}{3\sqrt{3}}\).

Step 1: Understanding the Cube Diagonal

The diagonal of a cube can be determined by using the Pythagorean theorem in 3 dimensions. The diagonal \(d\) of a cube relates to its edge length \(a\) by the formula \(d = a\sqrt{3}\). This represents the diagonal connecting two opposite vertices of the cube.

Step 2: Finding Edge Length as a Function of Diagonal

To express the edge length \(a\) as a function of the diagonal \(d\), we need to solve the equation \(d = a\sqrt{3}\) for \(a\). Thus, \(a = \frac{d}{\sqrt{3}}\).

Step 3: Expressing Surface Area as a Function of Diagonal

The surface area \(A\) of a cube with edge length \(a\) is given by \(A = 6a^2\). Substituting the expression for \(a\) from the previous step, the surface area as a function of the diagonal is \(A = 6 \left(\frac{d}{\sqrt{3}}\right)^2 = 2d^2\).

Step 4: Expressing Volume as a Function of Diagonal

The volume \(V\) of a cube with edge length \(a\) is \(V = a^3\). Substituting the expression for \(a\), the volume as a function of the diagonal is \(V = \left(\frac{d}{\sqrt{3}}\right)^3 = \frac{d^3}{3\sqrt{3}}\).

Key Concepts

Cube Diagonal

A cube diagonal stretches from one corner of the cube to the opposite corner, traversing the inside space. This diagonal is special because it does not lie on any face of the cube. To find the diagonal's length, we use the Pythagorean theorem in 3D: if each edge of the cube is of length \(a\), the diagonal \(d\) of the cube is given by the formula \(d = a\sqrt{3}\). This is derived because the diagonal forms a right-angled triangle with two other edges of the cube.
Solving for \(a\) when the diagonal length \(d\) is known involves isolating \(a\) in the equation. Thus, the edge length \(a\) is \(a = \frac{d}{\sqrt{3}}\). Knowing this relationship helps in understanding further properties like surface area and volume.

Surface Area of a Cube

The surface area of a cube is calculated by considering all six of its faces. Each face of a cube is a square, so if each edge of a cube is of length \(a\), the area of one face is \(a^2\). Since a cube has six such faces, the total surface area \(A\) is \(A = 6a^2\).
When given the diagonal length \(d\) instead of the edge length, we substitute \(a\) using the formula from the cube diagonal section. The expression for the surface area in terms of the diagonal becomes \(A = 6\left(\frac{d}{\sqrt{3}}\right)^2\). Simplifying this, we get \(A = 2d^2\). This formula shows how the surface area depends directly on the diagonal length squared.

Volume of a Cube

Volume measures how much space is enclosed within a cube. For a cube with an edge length of \(a\), the volume \(V\) is given by \(V = a^3\). It makes intuitive sense because a cube is essentially an extension of a square into three-dimensional space.
To express the volume in terms of the diagonal \(d\), we replace \(a\) with \(\frac{d}{\sqrt{3}}\). Thus, the volume becomes \(V = \left(\frac{d}{\sqrt{3}}\right)^3\). Simplifying yields the expression \(V = \frac{d^3}{3\sqrt{3}}\). This formula highlights how the volume changes with variations in the cube's diagonal length.