Problem 43 - Essential Calculus Early Transcendentals

Question

Find the angle between a diagonal of a cube and one of its edges.

Step-by-Step Solution

Answer

The angle is approximately 54.7 degrees.

Step 1: Identify the Components of the Cube

A cube has 6 faces, 12 edges, and 8 vertices. Each face of the cube is a square, and all edges are of equal length. We need to find the angle between a diagonal of the cube and one of its edges.

Step 2: Define the Edge Length and Diagonal

Assign a length to one edge of the cube, call it \( a \). A diagonal of the cube connects two opposite vertices through the interior and has a length. Use the Pythagorean theorem in three dimensions: \( d = \sqrt{a^2 + a^2 + a^2} = a\sqrt{3} \).

Step 3: Understand the Relevant Triangle

A right triangle can be formed with the edge of the cube, the diagonal of a face, and the diagonal of the cube itself. The edge is \( a \), the diagonal of the cube is \( a\sqrt{3} \), and the face diagonal is \( a\sqrt{2} \) (using the diagonal formula for a square).

Step 4: Use Cosine Rule for the Angle

Let \( \theta \) be the angle between the diagonal of the cube and one of its edges. Using the cosine rule, \( \,\cos(\theta) = \frac{a}{a\sqrt{3}}\).

Step 5: Calculate the Angle

Solve for \( \theta \): \( \,\cos(\theta) = \frac{1}{\sqrt{3}}\). Hence, \( \theta = \cos^{-1}(\frac{1}{\sqrt{3}}) \). This calculates to approximately \( 54.7\) degrees.

Key Concepts

Three-Dimensional Shapes

Three-dimensional shapes are objects that have length, width, and height. Unlike flat two-dimensional shapes, they exist in space and have volume. Understanding these shapes is fundamental in geometry, as they help us to comprehend the real-world structures around us.

• Length, Width, and Height: These three measurements define the size of a three-dimensional shape. For example, a box has a length, a width, and a height, making it clearly different from a flat rectangle, which has only length and width.
• Volume: This is a measure of the amount of space inside a three-dimensional shape. It is typically measured in cubic units.
• Surface Area: The total area of all the surfaces that make up the outside of the shape. For example, a cube has six equal square faces, so you would add all the areas of these squares to find the total surface area.

Recognizing and understanding three-dimensional shapes is an essential skill for solving problems involving real-life objects and structures.

Cube Geometry

A cube is a special type of three-dimensional shape that is both simple and fascinating. All the faces of a cube are squares, and all sides are equal in length. This symmetry makes it an excellent subject for geometric exploration.

• Properties of a Cube: A cube has six faces, each in the shape of a square. It has twelve edges, all the same length, and eight vertices, where three edges meet at each point.
• Diagonal in a Cube: A diagonal in a cube stretches from one corner to the opposite corner, passing through the interior space. This internal diagonal is longer than an edge and even longer than the diagonal of one of the cube’s square faces.
• Calculating Diagonals: The diagonal of a face of the cube is calculated using the formula for the diagonal of a square, which is \(a\sqrt{2}\), where \(a\) is the edge length of the cube. The diagonal of the cube, connecting opposite vertices, is \(a\sqrt{3}\).

Understanding cube geometry helps in solving problems related to spatial awareness and measurements.

Angle Calculation

Angle calculation can involve finding angles between various lines or planes. In this problem, the goal is to find the angle between a diagonal of a cube and one of its edges. This requires understanding both the concrete visualization and the mathematical principles.

• Right Triangles in Cubes: By forming a right triangle inside the cube, with the edge, face diagonal, and cube diagonal, you can apply geometric principles for solving angles. This makes use of the Pythagorean theorem in three dimensions.
• Using Trigonometric Functions: The cosine rule is particularly useful here. It expresses the cosine of an angle in a triangle in terms of the lengths of the sides. In our case, you can calculate \(\cos(\theta) = \frac{1}{\sqrt{3}}\).
• Solving the Angle: Once you have \(\cos(\theta)\), finding \(\theta\) involves using the inverse function, leading to \(\theta = \cos^{-1}(\frac{1}{\sqrt{3}})\), which is approximately 54.7 degrees.

Knowing how to calculate angles between geometric elements is critical for understanding and solving various spatial and architectural challenges.