Problem 20 - Algebra 2 and Trigonometry

Question

Ann and Bill Bekebrede follow a familiar triangular path when they take a walk. They walk mfrom home for 0.52 mile along a straight road, turn at an angle of \(95^{\circ},\) walk for another 0.46 mile, and then return home. a. Find, to the nearest hundredth of a mile, the length of the last portion of their walk. b. Find, to the nearest hundredth of a mile, the total distance that they walk.

Step-by-Step Solution

Answer

a. The last portion is approximately 0.71 miles. b. Total distance walked is 1.69 miles.

Step 1: Understanding the Triangle

The path described forms a triangle, with sides labeled as follows: \(a = 0.52\) miles, \(b = 0.46\) miles, and \(c\) (the side we need to find) is the return trip home. The angle between the first two sides is \(95^{\circ}\). We will use the Law of Cosines to find the unknown side \(c\).

Step 2: Applying the Law of Cosines

According to the Law of Cosines, the formula is \(c^2 = a^2 + b^2 - 2ab\cos(C)\). Substituting the given values: \(c^2 = 0.52^2 + 0.46^2 - 2 \times 0.52 \times 0.46 \times \cos(95^{\circ})\). Use a calculator to solve this.

Step 3: Calculating \(c\)

First, calculate the squares: \(0.52^2 = 0.2704\) and \(0.46^2 = 0.2116\). Next, calculate the cosine component: \(2 \times 0.52 \times 0.46 \times \cos(95^{\circ}) = -0.026404\) (using the cosine value for \(95^{\circ}\)). Substitute these into the formula: \(c^2 = 0.2704 + 0.2116 - (-0.026404)\). Simplify to get \(c^2 = 0.508404\). Taking the square root gives \(c \approx 0.71\) miles.

Step 4: Calculating the Total Distance

The total distance they walked is the sum of all sides of the triangle. This is \(0.52 + 0.46 + 0.71 \approx 1.69\) miles.

Key Concepts

Understanding Trigonometry in Triangles

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It's crucial in calculating distances and angles, especially when direct measurement is impossible.

The exercise from Ann and Bill's walking path is a perfect example of trigonometry in action. Here, we use the concept to define their triangular walking path, helping to find the unknown side.

One key formula in trigonometry for triangles is the Law of Cosines, which is specifically helpful when dealing with non-right triangles. This law extends the Pythagorean theorem and is expressed as: \[c^2 = a^2 + b^2 - 2ab\cos(C)\]where \(a\) and \(b\) are known sides and \(C\) is the angle between them. This allows calculation of the third side \(c\) when the triangle isn't a right triangle.

By understanding how trigonometry connects sides and angles, you can solve numerous real-world problems, like navigating paths or calculating distances between points.

The Importance of Triangles in Distance Calculation

Triangles play a pivotal role in distance calculation, especially in navigation and measurement tasks. The path of Ann and Bill is a triangle, showcasing how practical applications often form triangular patterns. Each of the triangle's sides and included angles contribute to understanding their walking path.

Each triangle type, from right-angled to obtuse, like this example with a \(95^{\circ}\) angle, offers various ways to calculate unknown elements such as distance. Identifying which type of triangle you're working with and the given measures helps determine the best trigonometric function or rule.

The Law of Cosines, for instance, is invaluable here because it applies to any triangle, not limited to right-angled ones. The formula assists in handling obtuse angles, ensuring accurate distance assessment without directly measuring the path. These principles show that triangles are foundational in mathematical and practical problem-solving strategies.

Accurate Distance Calculation Using the Law of Cosines

When tasked with calculating distances indirectly, the Law of Cosines is a critical tool. For Ann and Bill's triangle path, it calculates the distance of the unknown side, informing how far they've walked in total.

To apply the Law of Cosines effectively:

• Identify known triangle sides and angle (e.g., \(a = 0.52\) miles, \(b = 0.46\) miles, \(C = 95^{\circ}\)).
• Use cosine for the exact angle (\(C\)) involved. Trigonometric tables or calculators assist here.
• Substitute into the formula, solve for the unknown side, then calculate its square root for the final distance.

Calculating this way, Ann and Bill determine the final leg of their walk is approximately \(0.71\) miles, forming a complete loop. These steps also illustrate how mathematical tools transform abstract measures into tangible results, essential for real-world navigation.