Problem 50 - Algebra 2

Question

A person holds one end of a rope that runs through a pulley and has a weight attached to the other end. Assume the weight is directly beneath the pulley. The section of rope between the pulley and the weight is 12 feet long. The rope bends through an angle of 33 degrees in the pulley. How far is the person from the weight? F 7.8 ft G 10.5 ft H 12.9 ft J 14.3 ft

Step-by-Step Solution

Answer

The person is approximately 10.5 feet from the weight.

Step 1: Identify Known Values

We have a right triangle where the length of the rope from the pulley to the weight is the hypotenuse. The length of the rope is 12 feet and the angle given is 33 degrees.

Step 2: Determine the Triangle Configuration

Because the angle is given at the pulley, the distance from the person to the weight forms the base of a right triangle with the hypotenuse being 12 feet, and the opposite angle being 33 degrees.

Step 3: Apply the Cosine Function

The distance we need to find is adjacent to the angle of 33 degrees. We apply the cosine formula: \( \cos(\text{angle}) = \frac{\text{adjacent}}{\text{hypotenuse}} \).

Step 4: Calculate the Distance

Use the cosine function to find the distance. Plug in the values: \( \cos(33^{\circ}) = \frac{x}{12} \). Solve for \( x \): \( x = 12 \cdot \cos(33^{\circ}) \).

Step 5: Compute the Final Result

Calculate using a calculator: \( x = 12 \cdot 0.8387 \approx 10.06 \). The closest answer from the given options is 10.5 ft.

Key Concepts

Right Triangle

Right triangles are a fundamental aspect of geometry and trigonometry. These triangles have one angle equal to 90 degrees, known as the right angle. The side opposite this angle is the longest side, called the hypotenuse. The remaining sides are known as the adjacent and opposite sides. In our exercise, the rope forms a right triangle where:

• The hypotenuse is the 12-foot length of rope.
• The adjacent side is the distance from the person to the weight, the value we want to find.
• The opposite side would be the height of the weight below the pulley.

Understanding these components of a right triangle is crucial for applying trigonometric functions effectively and solving problems like the one at hand.

Cosine Function

The cosine function is a key tool in trigonometry. It relates the lengths of the sides of a right triangle to its angles. For an angle \(\theta\) in a right triangle, the cosine function is defined as the ratio of the length of the adjacent side to the hypotenuse:\[\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\]In the problem, the cosine function is used to find the distance (adjacent side) between the person and the weight. Given that the hypotenuse is 12 feet and the angle is 33 degrees, we can write the equation as:\[\cos(33^\circ) = \frac{x}{12}\]Solving this allows us to determine how far the person is from the weight.

Angle Measurement

Angle measurement is crucial in trigonometric calculations. Angles can be measured in degrees or radians, but for this exercise, we're using degrees. The angle in a right triangle is necessary to determine the relationships between sides using trigonometric functions. In this scenario, the given angle of 33 degrees at the pulley plays a significant role.

• It's important to note that angles in degrees can be converted to radians when needed, often using the formula \(\pi \, \text{radians} = 180^\circ\).
• Accurate measure of the angle helps in correctly applying the cosine function.

Using a protractor or a calculator with trigonometric functions can facilitate these measurements and ensure precision when solving trigonometry problems.

Solving Triangles

Solving triangles involves finding unknown angles or side lengths through known measures and trigonometric relationships. For right triangles, this typically involves functions such as sine, cosine, and tangent. In the exercise:

• The length of the hypotenuse and an angle were provided, facilitating the use of the cosine function to find the missing side.
• By rearranging the cosine formula, we directly solved for the adjacent side, giving us the distance between the person and the weight.
• This kind of problem-solving requires understanding which trigonometric function to apply based on the given and required information.

Trigonometry provides powerful techniques to solve real-world problems involving triangles, such as determining distances, angles, and configurations.