Problem 103 - A Graphical Approach to Precalculus with Limits

Question

Solve each application of operations and composition of functions. Area of an Equilateral Triangle The area \(a\) of an equilateral triangle with sides of length \(x\) is given by $$A(x)=\frac{\sqrt{3}}{4} x^{2}$$ (a) Find \(s l(2 x)\), the function representing the area of an equilateral triangle with sides of length twice the original length. (b) Find analytically the area of an equilateral triangle with side length \(16 .\) Use the given formula for \(s l(x)\) (c) Support the result of part (b) graphically.

Step-by-Step Solution

Answer

(a) \(A(2x) = \sqrt{3}x^2\); (b) Area is \(256\sqrt{3}\); (c) Confirm graphically by plotting \(A(x) = \sqrt{3}x^2\).

Step 1: Evaluate Function for Sides of Twice the Length

To find the area of an equilateral triangle with sides twice the original length, we need to substitute \(2x\) for \(x\) in the area formula. The original formula is \(A(x)=\frac{\sqrt{3}}{4} x^{2}\). So, for sides of length \(2x\), replace \(x\) with \(2x\): \[A(2x) = \frac{\sqrt{3}}{4} (2x)^2 = \frac{\sqrt{3}}{4} \cdot 4x^2 = \sqrt{3}x^2\]. Thus, the function representing the area is \(A(2x) = \sqrt{3}x^2\).

Step 2: Find the Area with Side Length 16

Using the derived function from Step 1, we substitute \(x = 16\) into \(A(x) = \sqrt{3}x^2\). Calculate: \[A(16) = \sqrt{3} \cdot (16)^2 = \sqrt{3} \cdot 256 = 256\sqrt{3}\]. This is the area of an equilateral triangle with side length 16.

Step 3: Graphical Support

To support the result graphically, plot the function \(A(x) = \sqrt{3}x^2\). On a graph, the x-axis represents the side lengths, and the y-axis represents the area. When \(x = 16\), the corresponding point should show the area as \(256\sqrt{3}\). This provides visual confirmation of the calculated area, as this point should lie on the curve defined by the function.

Key Concepts

Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides are equal in length. This equality also ensures that each of the internal angles is 60 degrees. Equilateral triangles are notably significant in geometry due to their high degree of symmetry. The formula for the area of an equilateral triangle, given its side length, simplifies many calculations. Understanding this shape is fundamental for solving problems that involve such triangles, especially when determining aspects like area, perimeter, and other related properties.

• All sides have equal length.
• All internal angles are 60 degrees.
• High symmetry makes calculation formulas simpler.

Their simple structure makes them a stepping stone to more complex geometric concepts. They appear in various mathematical contexts, including symmetry analysis and tessellation. Because of this, mastering equilateral triangles is essential for any student delving into geometry.

Area Calculation

Calculating the area of an equilateral triangle involves a specific formula that depends solely on the length of one of its sides. This formula is \[A(x)=\frac{\sqrt{3}}{4}x^2\]where \(x\) is the length of a side of the triangle. Let's break down how this formula works:

• The term \(x^2\) gives the square of the side length.
• Multiplying by \(\frac{\sqrt{3}}{4}\) adjusts the value to match the precise area specific to an equilateral triangle.

Using this formula allows easy calculation of the area for any equilateral triangle when you know the side length. For example, doubling the side length affects the area by a factor of 4, not just 2, since area is a square relationship. Hence, understanding how the elements of this formula interact is crucial for correctly finding the area in geometric problems.

Function Evaluation

When evaluating functions, we often aim to find the output for specific inputs. In the context of our equilateral triangle area formula, this involves substituting the side length directly into the function to get the area.For instance, let's evaluate the function for starting side length \(x\) doubled, \(2x\). Substitute \(2x\) into our area formula:

• \[A(2x) = \frac{\sqrt{3}}{4} (2x)^2 = \sqrt{3}x^2\]
• This function describes how the area changes when each side's length is twice the original.

Function evaluation is essential in mathematics since it provides a method to analyze how changes in input affect the output. This concept is particularly beneficial when examining geometric formula applications to see the impacts of scaling and other transformations.

Graphical Representation

Graphically representing a function helps to visually understand the mathematical relationship between variables—in this case, side length and area of an equilateral triangle.To illustrate this, plot the function \(A(x) = \sqrt{3}x^2\) on a graph, where the x-axis shows side lengths and the y-axis represents the area.

• At \(x = 16\), check the corresponding spot on the curve. It should have a y-value of \(256\sqrt{3}\).
• Such visualizations make it easier to confirm the calculated area for any given side length.

Studying the graph allows us to identify trends and patterns such as how rapidly the area grows with increasing side length. Creating graphs from functions is a valuable skill; it bridges the gap between numerical calculations and visual comprehension.