Problem 86 - Intermediate Algebra

Question

Suppose that we are given a cube with edges 12 centimeters in length. Find the length of a diagonal from a lower corner to the diagonally opposite upper corner. Express your answer to the nearest tenth of a centimeter.

Step-by-Step Solution

Answer

The diagonal is approximately 20.8 cm.

Step 1: Understanding the Cube's Diagonal

A cube has three equal edges meeting at each vertex. To find the length of a diagonal that stretches from one vertex across the cube to the opposite vertex, we can use the 3-dimensional Pythagorean theorem.

Step 2: Applying the 3D Pythagorean Theorem

Since each side of the cube is 12 cm, and a cube's internal diagonal can be viewed as the hypotenuse of a right triangle where the other two sides are two edges of the cube. Use the formula \( d^2 = a^2 + a^2 + a^2 \) for a cube's diagonal, where \( a \) is the side length.

Step 3: Calculating with the Values

Substitute \( a = 12 \) cm in the formula: \[ d^2 = 12^2 + 12^2 + 12^2 \] \[ d^2 = 3 \times 12^2 \] \[ d^2 = 3 \times 144 \] \[ d^2 = 432 \] Find \( d \) by calculating \( \sqrt{432} \).

Step 4: Finding the Square Root

Calculate \( \sqrt{432} \): \( \sqrt{432} \approx 20.8 \) when rounded to the nearest tenth.

Key Concepts

Cube Geometry

Understanding the geometry of a cube is fundamental to solving problems related to its structure. A cube is a three-dimensional shape that has six equal square faces, twelve equal edges, and eight vertices. Each of the length, width, and height of a cube are equal. In this exercise, each side of the cube is given as 12 cm. This uniformity of dimensions is what makes a cube distinct from other rectangular prisms.

To solve problems related to cube geometry, it is essential to understand how to calculate lengths and areas in three dimensions. For example:

• Volume of a cube: This is calculated by raising the side length to the third power, i.e., if side length is 'a', then Volume = \( a^3 \).
• Surface Area of a cube: This is found by calculating the area of one face and multiplying by six. So, Surface Area = \( 6a^2 \).

Understanding these basic properties makes it easier to approach problems involving cube diagonals.

Cube Diagonals

Diagonals in cubes can be both face diagonals and space diagonals. However, in this problem, we focus on the space diagonal, which runs from one corner of the cube to the opposite corner. This diagonal is not on the surface but passes through the inside of the cube.

To find the length of a space diagonal, we use the 3D Pythagorean theorem. This is an extension of the 2D Pythagorean theorem that you might be familiar with. The formula to find the diagonal of a cube with side length \( a \) is:

• \( d = \sqrt{3a^2} \)

This formula is derived from visualizing the space diagonal as the hypotenuse of a right triangle formed by three edges of the cube meeting at a vertex. In our case with a cube side of 12 cm, substituting into the formula gives:
* \( d = \sqrt{3 imes 12^2} = \sqrt{432} \)
The final calculation gives the diagonal length in centimeters.

Mathematical Problem Solving

Approaching mathematical problems, like finding the diagonal of a cube, involves clear steps and logical reasoning. Begin by identifying what is asked—in this case, finding the space diagonal of a cube.

Here's a simple approach to tackle such problems:

• Read the problem carefully and identify the known quantities and what needs to be found.
• Choose the appropriate formula or theorem based on the geometry involved. In this case, the 3D Pythagorean theorem fits perfectly.
• Substitute the known values into the formula. Double-check your inputs for accuracy.
• Solve step-by-step, simplify your expressions where possible.
• Round the final answer as specified, ensuring units such as centimeters are included.

This methodical approach helps in solving not just cube-related problems, but a wide range of mathematical problems effectively. Bringing clarity to each step ensures a deeper understanding and confidence in problem solving.