/*! 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 57 To find the distance \(A B\) acr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 57

To find the distance \(A B\) across a river, a distance \(B C=354\) meters is laid off on one side of the river. It is found that \(B=112^{\circ} 10^{\prime}\) and \(C=15^{\circ} 20^{\prime} .\) Find \(A B\)

Short Answer

Expert verified
The distance across the river, \(AB\), is approximately 117.92 meters.

Step by step solution

01

Identify the Known Angles

We are given two angles: \(B = 112^\circ 10^\prime\) and \(C = 15^\circ 20^\prime\). To find angle \(A\), we use the fact that the sum of angles in a triangle is \(180^\circ\). Thus, \(A = 180^\circ - B - C = 180^\circ - 112^\circ 10^\prime - 15^\circ 20^\prime\).
02

Calculate Angle A

Subtract the given angles from \(180^\circ\) to find \(A\):\[A = 180^\circ - 112^\circ 10^\prime - 15^\circ 20^\prime = 52^\circ 30^\prime.\]
03

Application of the Law of Sines

Using the Law of Sines, \( \frac{AB}{\sin C} = \frac{BC}{\sin A} \). Rearrange the formula to find \(AB\):\[AB = BC \cdot \frac{\sin C}{\sin A}.\]
04

Calculate AB Using Sine Values

Given \(BC = 354\) meters, calculate \(AB\): \[AB = 354 \cdot \frac{\sin(15^\circ 20^\prime)}{\sin(52^\circ 30^\prime)}.\]Use a calculator to find \(\sin(15^\circ 20^\prime) \approx 0.264\) and \(\sin(52^\circ 30^\prime) \approx 0.793\). Substitute these values:\[AB = 354 \cdot \frac{0.264}{0.793} \approx 117.92 \text{ meters}.\]
05

Conclusion

The calculated distance across the river from point \(A\) to point \(B\) is approximately \(117.92\) meters.

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.

Angle Calculation
Calculating angles in a triangle is crucial in many geometric problems. In this context, we are dealing with angles within a triangle formed to measure a distance across a river. Triangles always have three angles, and the sum of these angles is always exactly 180 degrees. This property is fundamental to solving many triangle-related problems and makes finding an unknown angle straightforward.

For example, if you have two angles of a triangle, you can find the third one by subtracting the known angles from 180 degrees. Let's apply this to the given problem:
  • Angle B is given as 112 degrees and 10 minutes.
  • Angle C is 15 degrees and 20 minutes.
  • To find angle A, subtract these from 180 degrees: \[ A = 180^\circ - (112^\circ 10') - (15^\circ 20') = 52^\circ 30' \]
Breaking angles into degrees and minutes is common and sometimes necessary for precise angle measurements. Make sure to consider these units when calculating.
Triangle Properties
Understanding triangle properties helps in solving geometric problems more effectively. There are several basic properties of triangles that one should keep in mind:
  • The sum of the internal angles is always 180 degrees.
  • The longest side is opposite the largest angle.
  • The Law of Sines relates the lengths of the sides to the sines of their opposite angles.
In our scenario, we're primarily using the Law of Sines to determine the unknown distance across the river. This law states:

\[ \frac{AB}{\sin C} = \frac{BC}{\sin A} \]

By rearranging this formula, you can solve for side AB, provided you have potential values for angles and one side. Having a clear picture of these properties aids in understanding how to set up and rearrange equations to find unknown values.
Distance Measurement
The main objective here is to measure the distance across a river using properties of triangles. To achieve this, a practical application of the Law of Sines is employed. Here, we use the given side length and angles to find the unknown side, which is the distance from point A to point B.

The process involves these steps:
  • Identify the known side, which in this case is \( BC = 354 \) meters.
  • Use the Law of Sines to set up the equation: \[ AB = 354 \cdot \frac{\sin(15^\circ 20')}{\sin(52^\circ 30')} \]
  • Calculate the sine values using a calculator:
    • \( \sin(15^\circ 20') \approx 0.264 \)
    • \( \sin(52^\circ 30') \approx 0.793 \)
  • Substitute these values back into the formula and solve for AB: \[ AB = 354 \cdot \frac{0.264}{0.793} \approx 117.92 \text{ meters} \]
Remember, accurate calculations and understanding the relationships between sides and angles are key to correctly measuring the distance.

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

Radio stations sometimes do not broadcast in all directions with the same intensity. To avoid interference with an existing station to the north, a new station may be licensed to broadcast only east and west. To create an east- west signal, two radio towers are used, as illustrated in the figure. (IMAGE CAN'T COPY). Locations where the radio signal is received correspond to the interior of the lemniscate..$$r^{2}=40,000 \cos 2 \theta$$.Where the polar axis (or positive \(x\) -axis) points east. (A) Graph $$r^{2}=40.000 \cos 2 \theta$$. For \(0^{\circ} \leq \theta \leq 180^{\circ},\) with units in miles. Assuming that the radio towers are located near the pole, use the graph to describe the regions where the signal can be received and where the signal cannot be received. (B) Suppose a radio signal pattern is given by $$r^{2}=22,500 \sin 2 \theta$$ for \(0^{\circ} \leq \theta \leq 180^{\circ} .\) Graph this pattem and interpret the results.

Do the following. (a) Determine the parametric equations that model the path of the projectile. (b) Determine the rectangular equation that models the path of the projectile. (c) Determine the time the projectile is in flight and the horizontal distance covered. A model rocket is launched from the ground with a velocity of 48 feet per second at an angle of \(60^{\circ}\) with respect to the ground.

Find each quotient and express it in rectangular form by first converting the numerator and the denominator to trigonometric form. $$\frac{-i}{1+i}$$

Explain the condition that must exist to determine that there is no triangle satisfying the given values of \(a, b,\) and \(B\) once the value of \(\sin A\) is found.

The screen shown to the right is an example of a Lissajous figure. Lissajous figures occur in electronics and may be used to find the frequency of an unknown voltage. Graph each Lissajous figure for \(0 \leq t \leq 6.5\) in the window \([-6.6,6.6]\) by \([-4.1,4.1]\). (GRAPH CANNOT COPY) $$x=2 \sin 2 t, y=3 \cos 3 t$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

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.