Problem 61 - Intermediate Algebra

Question

The sides of a regular hexagonal nut are 10 millimeters long. Find the height \(h\) of the nut. Give the exact answer and an approximation to two decimal places. (IMAGE CANT COPY)

Step-by-Step Solution

Answer

The exact height of the nut is \( 5\sqrt{3} \) millimeters, approximately 8.66 millimeters.

Step 1: Understand the Regular Hexagon

A regular hexagon can be divided into 6 equilateral triangles, each with side length equal to the side of the hexagon. In this problem, each side of these triangles is 10 millimeters. To find the height of the hexagon, we need the height of one of these equilateral triangles.

Step 2: Find the Height of an Equilateral Triangle

For an equilateral triangle with side length \( s \), the height \( h \) can be found using the formula: \[ h = \frac{\sqrt{3}}{2} \times s \] Substituting \( s = 10 \) millimeters,\[ h = \frac{\sqrt{3}}{2} \times 10 \]

Step 3: Calculate the Exact Height

Calculate the exact height using the previous formula:\[ h = \frac{\sqrt{3}}{2} \times 10 = 5\sqrt{3} \text{ millimeters} \].This is the exact height of the hexagon.

Step 4: Approximation to Two Decimal Places

To find an approximate value for \( h \), calculate \( 5\sqrt{3} \) using a calculator:\[ 5\sqrt{3} \approx 5 \times 1.732 = 8.66 \text{ millimeters} \].Thus, the approximate height is 8.66 millimeters.

Key Concepts

Understanding a Regular Hexagon

A regular hexagon is a geometric shape with six equal sides and angles. This makes all its interior angles equal to 120 degrees. It is a symmetrical figure, which can be evenly divided into smaller, congruent shapes. A key property of a regular hexagon is that it can be divided into six equilateral triangles. Each triangle has the same side lengths as the side of the hexagon itself.
This characteristic is crucial when you need to calculate certain features, such as the height of the hexagon. By focusing on these equilateral triangle divisions, you can derive useful geometric parameters.
In our specific exercise, the hexagon sides are each 10 millimeters. This length becomes the side length of each equilateral triangle within the hexagon. Knowing this allows us to easily use trigonometric and geometric formulas to find dimensions like height or area.

Properties of an Equilateral Triangle

An equilateral triangle has all three sides of equal length, and all three angles are equal to 60 degrees. This symmetry makes calculations involving equilateral triangles straightforward. The formula for the height of an equilateral triangle is derived from basic trigonometric principles.
To find the height, you divide the equilateral triangle into two right triangles, with the height acting as one of the legs. This splits the base of the equilateral triangle into two equal halves. Using the Pythagorean theorem or the sine function, you can derive the height formula:

• The formula is: \[ h = \frac{\sqrt{3}}{2} \times s \] where \( s \) is the side length.

In our exercise, with side length 10 millimeters, substituting into the formula gives us a height of 5\( \sqrt{3} \) millimeters. Understanding these properties of equilateral triangles is central to solving geometric problems involving hexagons.

Height Calculation in Practice

Calculating the height of the hexagon involves accurately using the formula derived from the properties of equilateral triangles. Here, the side length value, given as 10 millimeters, directly impacts the outcome.

• Using the formula: \[ h = \frac{\sqrt{3}}{2} \times 10 = 5\sqrt{3} \] millimeters.
• This represents the exact height of the hexagon.
• An approximation involves multiplying \( 5\sqrt{3} \) by an approximate value for \( \sqrt{3} \), which is approximately 1.732.
• This calculation \[ 5 \times 1.732 = 8.66 \] millimeters provides a more practical, decimal-composed number.

Learning how to switch between exact values and decimal approximations is essential in dealing with real-world applications, ensuring precision without sacrificing everyday usability.