Problem 13 - Precalculus

Question

A building contractor wants to put a fence around the perimeter of a flat lot that has the shape of a right triangle. One angle of the triangle is \(41.4^{\circ},\) and the length of the hypotenuse is \(58.5 \mathrm{m}\). Find the length of fencing required. Round the answer to one decimal place.

Step-by-Step Solution

Answer

The length of fencing required is 141.0 m.

Step 1: Understand the Triangle

We have a right triangle where one angle is given as \(41.4^{\circ}\) and the hypotenuse is \(58.5\, \text{m}\). Using this information, we need to find the lengths of the other two sides in order to calculate the perimeter.

Step 2: Use the Sine Function to Find One Side

For the angle \(\theta = 41.4^{\circ}\), we can use the sine function to find the length of the opposite side, \(a\). The formula is: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]Substituting the known values, we have: \[ \sin(41.4^{\circ}) = \frac{a}{58.5} \]Solving for \(a\), we find: \[ a = 58.5 \cdot \sin(41.4^{\circ}) \approx 38.6 \text{m}\]

Step 3: Use the Cosine Function to Find Another Side

Next, we use the cosine function to find the adjacent side, \(b\): \[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]Substituting the known values: \[ \cos(41.4^{\circ}) = \frac{b}{58.5} \]Solving for \(b\), we find: \[ b = 58.5 \cdot \cos(41.4^{\circ}) \approx 43.9 \text{m}\]

Step 4: Calculating the Perimeter

Now that we have the lengths of all sides, we can calculate the perimeter of the triangle, which is the sum of all its sides: \[ P = a + b + \text{hypotenuse} = 38.6 + 43.9 + 58.5 \]Calculating this gives us: \[ P = 141.0 \text{m} \]

Step 5: Round and Present the Result

The final answer is already in one decimal place, so no further rounding is necessary. The total length of fencing required is \(141.0\, \text{m}\).

Key Concepts

Right Triangle

When dealing with a right triangle, one of its angles is always exactly 90 degrees. This fixed angle differentiates it from other types of triangles. Understanding the properties of a right triangle is key. Since one angle is 90 degrees, the sum of the other two angles equals 90 degrees. Hence, they are complementary. In any right triangle, the side opposite the right angle is the longest and is known as the hypotenuse. The other two sides are referred to as the adjacent and opposite sides, relative to the angle in question.
Right triangles are fundamental in trigonometry, as they are often used to explore and apply trigonometric ratios and functions, such as sine, cosine, and tangent.

Sine and Cosine Functions

The sine and cosine functions are essential tools in trigonometry, often used when dealing with right triangles. These functions help us determine unknown side lengths when one side and one angle are known.
For a given angle in a right triangle:

• The sine function is defined as the ratio of the length of the opposite side to the hypotenuse. Mathematically, it is expressed as \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).
• The cosine function is the ratio of the length of the adjacent side to the hypotenuse, given by \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \).

These functions depend on the angle measured, making them unique for each specific angle. Using these ratios, trigonometric calculations allow us to find unknown sides with ease in problems involving right triangles.

Perimeter Calculation

The perimeter of a geometry shape refers to the total distance around the shape. In the case of a right triangle, it involves summing up the lengths of all three sides: the two legs (opposite and adjacent) and the hypotenuse.
Given all side lengths, the perimeter \( P \) can be calculated using the formula:

• \( P = \text{opposite} + \text{adjacent} + \text{hypotenuse} \)

To compute the perimeter, one might need to use trigonometric functions if all side lengths are not directly known. For instance, knowing an angle and a hypotenuse, sine and cosine can be used to derive the opposite and adjacent sides, respectively. Once all sides are known, adding them gives the perimeter, as shown in the solution example. This basic principle is fundamental when solving perimeter-related problems.

Angle Measurement

Angle measurement is crucial in problems involving right triangles, as it can influence calculations by determining the relative lengths of triangle sides. Trigonometry revolves around understanding these angles.
Angles in mathematical problems are often measured in degrees, with a full circle totaling 360 degrees. For right triangles, attention is often focused on the angles other than the right angle (90 degrees), which are necessary for applying functions like sine and cosine.
In the example given, an angle is specified as \( 41.4^{\circ} \), leaving the other non-right angle to be \( 48.6^{\circ} \) because the angles in a triangle sum up to 180 degrees. Detailed knowledge of angles aids in employing trigonometric functions effectively, allowing us to solve for the unknown quantities reliably.