Problem 23 - Algebra 2 and Trigonometry

Question

Two sides of a triangular lot form angles that measure \(29.1^{\circ}\) and \(33.7^{\circ}\) with the third side, which is 487 feet long. To the nearest dollar, how much will it cost to fence the lot if the fencing costs \(\$ 5.59\) per foot?

Step-by-Step Solution

Answer

The cost will be approximately \( \$ \text{1,633} \).

Step 1: Understand the Problem

We have a triangular lot with angles measuring \( 29.1^{\circ} \) and \( 33.7^{\circ} \) adjacent to the third side. This third side is 487 feet long. We need to find the lengths of the other two sides to determine the perimeter of the triangle and calculate the cost of enclosing the lot with fencing.

Step 2: Calculate the Third Angle

The sum of angles in a triangle is always \(180^{\circ}\). Therefore, the third angle can be calculated as:\[180^{\circ} - 29.1^{\circ} - 33.7^{\circ} = 117.2^{\circ}\]

Step 3: Use the Law of Sines

To find the lengths of the other two sides, we use the Law of Sines: \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]Where \(a, b, c\) are sides opposite angles \(A, B, C\), respectively. Let side \(a\) be opposite \(29.1^{\circ}\), side \(b\) be opposite \(33.7^{\circ}\), and side \(c = 487\) be opposite \(117.2^{\circ}\).

Step 4: Find Side a

Using the Law of Sines for \(a\):\[\frac{a}{\sin 29.1^{\circ}} = \frac{487}{\sin 117.2^{\circ}}\]Solving for \(a\):\[a = \frac{487 \times \sin 29.1^{\circ}}{\sin 117.2^{\circ}}\]Use a calculator to find \(a\).

Step 5: Find Side b

Using the Law of Sines for \(b\):\[\frac{b}{\sin 33.7^{\circ}} = \frac{487}{\sin 117.2^{\circ}}\]Solving for \(b\):\[b = \frac{487 \times \sin 33.7^{\circ}}{\sin 117.2^{\circ}}\]Use a calculator to find \(b\).

Step 6: Calculate the Perimeter

Add up the sides to find the perimeter:\[ \text{Perimeter} = a + b + c\]Where \(a\) and \(b\) are found from the previous steps, and \(c = 487\).

Step 7: Calculate the Cost

Multiplying the perimeter by the cost of fencing per foot gives the total cost:\[\text{Cost} = \text{Perimeter} \times 5.59\]Round the result to the nearest dollar.

Key Concepts

Triangle Geometry

Understanding triangle geometry is essential when solving problems involving triangles, just like in the given exercise. A triangle is a three-sided polygon, and in geometry, it is often defined by its sides, angles, and the relationships between them. Here, we have a triangular lot with two given angles and a known side length, and our task is to find the missing side lengths and perimeter.

The basic components of a triangle can be summarized as follows:

• Three sides: Often denoted as \(a\), \(b\), and \(c\).
• Three angles: Typically corresponding to the sides they face, noted as \(A\), \(B\), and \(C\).

Triangles can be categorized based on side lengths and angles, such as scalene, isosceles, and right triangles, but in this exercise, we are dealing with an obtuse triangle because of the angle greater than 90 degrees. Understanding the basic triangle geometry concepts helps to apply mathematical formulas correctly, such as using the Law of Sines to find unknown side lengths.

Angle Sum Property

The angle sum property of a triangle states that the sum of all interior angles in any triangle is always \(180^{\circ}\). This is one of the fundamental properties of triangles used in geometry. In the given problem, you need to calculate the missing angle of the triangular lot to use it effectively in further calculations.

Given the angles \(29.1^{\circ}\) and \(33.7^{\circ}\), the angle sum property allows us to find the third angle by applying the formula:

\[ 180^{\circ} - 29.1^{\circ} - 33.7^{\circ} = 117.2^{\circ} \]

This property is always consistent and provides a basis for solving various triangle problems. Even if one of the angles is large, as it is here, the property holds true, and ensures the triangle is correctly constructed and analyzed.

Perimeter Calculation

Calculating the perimeter of a triangle involves adding the lengths of all three sides together. To determine the perimeter in our exercise, we first need to find all sides using the Law of Sines. Once you have the side lengths, perimeter calculation becomes straightforward. Here’s how you execute the steps:

1. First, determine the lengths of sides \(a\) and \(b\) using:

• \[ a = \frac{487 \times \sin 29.1^{\circ}}{\sin 117.2^{\circ}} \]
• \[ b = \frac{487 \times \sin 33.7^{\circ}}{\sin 117.2^{\circ}} \]

2. The provided side, \(c\), is already known as 487 feet.
3. Add the side lengths to find the total perimeter:

• \[ \text{Perimeter} = a + b + c \]

Calculating the perimeter is crucial as it gives a measure of the boundary around the triangle, which is necessary for determining fencing costs.

Cost Estimation

Once the perimeter is known, estimating the cost of fencing the triangular lot can be straightforward. The idea is to multiply the perimeter by the cost per foot of fencing, given as \( \$5.59 \) per foot in this problem. Here’s a simple formula you can use:

\[ \text{Cost} = \text{Perimeter} \times 5.59 \]

Let’s break down the cost estimation steps:

• First, calculate the perimeter using the steps from the previous section.
• Next, multiply this perimeter by the cost per foot.
• Finally, round the total to the nearest whole dollar to get the cost estimation.

This provides an easy way to determine the total cost required for fencing, ensuring you stay within budget by checking the nearest round figure to plan for expenses efficiently.