Problem 22 - Algebra and Trigonometry with Analytic Geometry

Question

Surveying Two points \(P\) and \(Q\) on level ground are on opposite sides of a building. To find the distance between the points, a surveyor chooses a point \(R\) that is 300 feet from \(P\) and 438 feet from \(Q\) and then determines that angle \(P R Q\) has measure \(37^{\circ} 40^{\prime}\) (see the figure). Approximate the distance between \(P\) and \(Q\).

Step-by-Step Solution

Answer

The distance between points \(P\) and \(Q\) is approximately 272 feet.

Step 1: Understand the Problem

We are tasked with finding the distance between two points, \(P\) and \(Q\), on opposite sides of a building. The surveyor has chosen a point \(R\) and measured the distances \(PR = 300\) feet and \(QR = 438\) feet. The angle \(\angle PRQ = 37^{\circ} 40^{\prime}\). We need to use these measurements to find \(PQ\).

Step 2: Convert Angle Measurement

First, convert the angle measurement from degrees and minutes to a decimal value. Since there are 60 minutes in a degree, we convert \(37^{\circ} 40^{\prime}\) as follows: \(37 + \frac{40}{60}\). This gives us \(37.6667^{\circ}\).

Step 3: Apply the Law of Cosines

The Law of Cosines states that for a triangle with sides \(a\), \(b\), and \(c\), opposite to angles \(A\), \(B\), and \(C\) respectively, the following is true: \[ c^2 = a^2 + b^2 - 2ab \cos(C) \]In this problem, let \(a = 300\), \(b = 438\), and \(C = 37.6667^{\circ}\), where \(c\) represents the distance \(PQ\).

Step 4: Calculate Using Law of Cosines

Substitute the known values into the Law of Cosines formula to find \(c\):\[ c^2 = 300^2 + 438^2 - 2 \times 300 \times 438 \times \cos(37.6667^{\circ}) \]First, calculate \(\cos(37.6667^{\circ}) \approx 0.7933\).

Step 5: Solve for Distance

Substitute \(\cos(37.6667^{\circ})\) into the equation:\[ c^2 = 90000 + 192144 - 2 \times 300 \times 438 \times 0.7933 \]Simplify and solve for \(c\):\[ c^2 = 90000 + 192144 - 208113.6 \]\[ c^2 = 74030.4 \]\[ c = \sqrt{74030.4} \approx 271.98 \]Thus, the distance \(PQ\) is approximately 271.98 feet.

Key Concepts

Surveying

Surveying is a crucial part of mapping and construction, as it helps determine and visualize the layout of land. In our exercise, the surveyor used basic principles of geometry to resolve real-world spatial questions. Let's break down how surveying was used in this scenario to determine the distance between points that aren't directly measurable.

The surveyor identified a third point, labeled as \( R \), placed strategically with known distances from \( P \) and \( Q \). Then, through measurement and calculation, the surveyor could compute the unknown distance \( PQ \) by utilizing the Law of Cosines.

• Determining points: It involves placing a reference point that makes complex measurements possible.
• Measuring distances: Direct measurements are taken from this reference to each point of interest.
• Using angles: Known angles between different points facilitate calculating the unknown distances.

Surveying relies strongly on geometry and trigonometry, making it possible to solve problems involving inaccessible distances, just like finding the hidden distance across a building in our problem.

Angle Conversion

Understanding angle conversion is essential when dealing with real-world mathematical problems, especially in surveying and navigation where precise calculations are necessary. In this exercise, the angle \( \angle PRQ \) was given in degrees and minutes, and converting it into a decimal form was a pivotal step to apply trigonometric functions effectively.

Here is a simple overview of angle conversion:

• Angles are often expressed in degrees, minutes (\

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