Problem 70 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Distance between Ends of the Vietnam Memorial The Vietnam Veterans Memorial in Washington, DC, is V-shaped with equal sides of length 246.75 feet, and the angle between these sides measures \(125^{\circ} 12^{\prime} .\) Find the distance between the ends of the two sides. (Source: Pamphlet obtained at Vietnam Veterans Memorial.) (picture cannot copy)

Step-by-Step Solution

Answer

The distance is approximately 437.67 feet.

Step 1: Understand the Problem

We are given a V-shaped structure with two sides each measuring 246.75 feet. The angle between these sides is given as \(125^{\circ} 12^{\prime}\). We are tasked with finding the distance between the ends of these two sides. This is a classic application of the law of cosines.

Step 2: Convert Angle to Decimal Degrees

The angle is given in degrees and minutes. First, we convert it into decimal degrees. There are 60 minutes in a degree, so \(12\) minutes is \(12/60 = 0.2\) degrees. Thus, the angle is \(125.2^{\circ}\).

Step 3: Recall the Law of Cosines Formula

The law of cosines is used when we have two sides and the included angle, or we have three sides. The formula is \( c^2 = a^2 + b^2 - 2ab\cos(C) \), where \( c \) is the side opposite angle \( C \), and \( a \) and \( b \) are the other two sides.

Step 4: Plug Values into the Formula and Calculate

We replace \( a \) and \( b \) in the cosine formula with our side lengths, and use \( C = 125.2^{\circ} \). This gives us:\[ c^2 = 246.75^2 + 246.75^2 - 2 \times 246.75 \times 246.75 \times \cos(125.2^{\circ}) \]Calculate \( c \) by first evaluating each part of the equation.

Step 5: Calculate the Distance

After calculating the values:\[ 246.75^2 = 60889.5625 \]The expression becomes:\[ c^2 = 60889.5625 + 60889.5625 - 2 \times 246.75 \times 246.75 \times (-0.5735) \]After simplifying, we calculate:\[ c^2 = 60889.5625 + 60889.5625 + 69879.2366 \]\[ c = \sqrt{191658.3616} \approx 437.67 \text{ feet} \]

Step 6: Round Off to Appropriate Significant Figures

Since the lengths are given to two decimal places, the computed distance should also be rounded to two decimal places. Thus, the distance between the ends of the two sides is approximately 437.67 feet.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. One of the key tools used in trigonometry is the law of cosines, which helps us find a missing side or angle in any triangle, especially non-right triangles. In our problem, the Vietnam Veterans Memorial forms a V-shaped triangle, where we need to find the length of the side opposite a known angle. The law of cosines is essential in such scenarios because unlike the Pythagorean Theorem, it applies to all types of triangles:

• It provides a way to calculate unknown lengths based on known sides and angles.
• It is particularly useful when dealing with oblique triangles, which do not have a right angle.
• The formula can be rearranged to solve for angles if all side lengths are known.

Understanding these principles helps us systematically tackle the distance calculation between the ends of the memorial.

Angle Conversion

In many geometry problems, angles are given in mixed notation of degrees and minutes. It’s crucial to convert these angles into decimal degrees for mathematical calculations. This involves understanding that:

• There are 60 minutes in one degree. Thus, to convert minutes to degrees, you divide the minutes by 60.
• By converting to decimal degrees, we can more easily use trigonometric functions such as cosine.

In the given problem, the angle between the two equal sides of the V-shaped memorial is \(125^{\circ} 12^{\prime}\). By converting, we found that this equals \(125.2^{\circ}\). This conversion is crucial for accurately applying the law of cosines and carrying out any needed trigonometric operations.

Distance Calculation

Once the angle is converted to decimal degrees and with both sides known, we can apply the law of cosines to solve for the missing side. The formula used is:\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]where \(a\) and \(b\) are the legs of the triangle, and \(C\) is the angle between them. The steps involve:

• Inserting the given lengths and angle into the formula.
• Calculating each component properly, like squaring the side lengths and finding the cosine of the angle.
• Simplifying the terms to get \(c^2\) and finally taking the square root.

This process helps in determining the distance between the ends of the two sides of the memorial accurately.

Geometry Problem Solving

Solving geometry problems requires a blend of analytical thinking and practical application of formulas like the law of cosines. The task with the Vietnam Memorial involves:

• Understanding the geometric shape and determining the type of triangle formed.
• Identifying known versus unknown elements in the problem.
• Sequentially using trigonometric principles to compute the desired measurement.

By breaking down the problem into manageable steps such as converting angles, applying the correct formula, and systematically solving for the unknown, the problem becomes much more approachable. This methodical approach ensures an accurate and reliable solution, which is critical for making sense of complex geometry problems.