/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 54 A 100 -ft guy wire is attached t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 54

A 100 -ft guy wire is attached to the top of an antenna. The angle between the guy wire and the ground is \(62^{\circ} .\) How tall is the antenna (to the nearest foot)?

Short Answer

Expert verified
The height of the antenna is approximately 88 feet.

Step by step solution

01

Understand the problem

The problem involves a right triangle where the hypotenuse is the guy wire, the angle with the ground is given, and the height of the antenna is the side opposite to this angle. Use trigonometry to find the height.
02

Identify known values

The length of the guy wire (hypotenuse) is 100 feet, and the angle between the guy wire and the ground is 62 degrees.
03

Set up the trigonometric function

Use the sine function, which relates the opposite side (height of the antenna) to the hypotenuse in a right triangle. The formula is \(\text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\).
04

Solve for the height

Substitute the known values into the equation: \(\text{sin}(62^\text{circ}) = \frac{h}{100}\).
05

Calculate the value

Rearrange to solve for the height \(\text{h} = 100 \times \text{sin}(62^\text{circ})\). Use a calculator to find \(\text{sin}(62^\text{circ})\), which is approximately 0.8829.
06

Final computation

Multiply: \(\text{h} = 100 \times 0.8829 \approx 88\). Therefore, the height of the antenna is approximately 88 feet.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Right Triangle
A right triangle is a triangle that has one angle measuring exactly 90 degrees. The sides of a right triangle include the hypotenuse, the opposite side, and the adjacent side:
  • Hypotenuse: This is the longest side and is opposite the right angle.
  • Opposite side: This is the side opposite the angle of interest in trigonometric calculations.
  • Adjacent side: This is the side that forms the angle of interest along with the hypotenuse.
These properties make the right triangle a common subject in trigonometry problems.
Trigonometric Functions
Trigonometric functions are essential tools in solving problems related to angles and distances in triangles. They are based on the relationship between the angles and sides of a right triangle. The three primary trigonometric functions are:
  • Sine (sin): It is the ratio of the opposite side to the hypotenuse. Formulated as \(\text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).
  • Cosine (cos): It is the ratio of the adjacent side to the hypotenuse. Formulated as \(\text{cos}(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \).
  • Tangent (tan): It is the ratio of the opposite side to the adjacent side. Formulated as \(\text{tan}(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
These functions can be used to calculate missing sides or angles in right triangles.
Sine Function
The sine function specifically helps in determining the relationship between a given angle and the length of the sides in a right triangle. When dealing with a given angle and looking to find the opposite side or the hypotenuse, the sine function is useful because:
  • It provides a direct ratio of the opposite side to the hypotenuse.
  • It assists in solving real-world problems like figuring out heights and distances when an angle and hypotenuse are known.
Using the formula \(\text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \), you can calculate unknown values by rearranging the equation. For instance, to find the height of an antenna with a guy wire of 100 feet at a 62-degree angle to the ground, you would set up the equation \(\text{sin}(62^\text{circ}) = \frac{h}{100} \). Rearranging results in \(\text{h} = 100 \times \text{sin}(62^\text{circ}) \).
Angle Measurement
Angle measurement is crucial in solving trigonometric problems. Angles can be measured in degrees or radians, but degrees are often used in practical applications like this exercise. Key points to remember about angle measurement include:
  • Degrees: A full revolution is 360 degrees. Angles are commonly expressed in degrees.
  • Using a calculator: Ensure the calculator is set to the correct mode (degrees) when working with angle problems.
  • Conversion: For certain calculations, you might need to convert between degrees and radians, though this exercise uses degrees only.
By understanding how to measure and use angles, you can accurately apply trigonometric functions to find unknown sides or angles in a triangle.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find each product. Be sure to indicate the units for the answer. Round approximate answers to the nearest tenth. $$ \frac{300 \mathrm{rad}}{1 \mathrm{hr}} \cdot \frac{1 \mathrm{hr}}{60 \mathrm{~min}} $$

Convert each angle to degrees-minutes-seconds. Round to the nearest whole number of seconds. $$ -17.33^{\circ} $$

A 41 -m guy wire is attached to the top of a 34.6-m antenna and to a point on the ground. How far is the point on the ground from the base of the antenna (to the nearest meter), and what angle does the guy wire make with the ground (to the nearest tenth of a degree)?

Evaluate each expression without using a calculator. Give the result in degrees. \(\cos ^{-1}\left(\frac{1}{2}\right)\)

Eratosthenes Measures Earth Over 2200 years ago Eratosthenes read in the Alexandria library that at noon on June 21 a vertical stick in Syene cast no shadow. So on June 21 at noon Eratosthenes set out a vertical stick in Alexandria and found an angle of \(7^{\circ}\) in the position shown in the drawing. Eratosthenes reasoned that since the sun is so far away, sunlight must be arriving at Earth in parallel rays. With this assumption he concluded that Earth is round and the central angle in the drawing must also be \(7^{\circ} .\) He then paid a man to pace off the distance between Syene and Alexandria and found it to be \(800 \mathrm{~km}\). From these facts, calculate the circumference of Earth (to the nearest kilometer) as Eratosthenes did and compare his answer with the circumference calculated by using the currently accepted radius of \(6378 \mathrm{~km}\).
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.