/*! 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 63 From the top of a 250 -foot ligh... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 63

From the top of a 250 -foot lighthouse, a plane is sighted overhead and a ship is observed directly below the plane. The angle of elevation of the plane is \(22^{\circ}\) and the angle of depression of the ship is \(35^{\circ}\). Find a. the distance of the ship from the lighthouse; b. the plane's height above the water. Round to the nearest foot.

Short Answer

Expert verified
a. The distance of the ship from the lighthouse is approximately 366ft. b. The total height of the plane from the water is approximately 1043ft.

Step by step solution

01

Establish the Scenario

First Identify all given information and visualize the scenario. From the top of a lighthouse, two right triangles can be formed using the line of sight to the plane and the ship. The angle of elevation to the plane is \(22^{\circ}\) and the angle of depression to the ship is \(35^{\circ}\). This means that the angle of depression to the plane from the horizontal is \(90^{\circ}-22^{\circ}=68^{\circ}\). Since depression and elevation are opposite, they are equal, so the angle of elevation from the ship to the plane is also \(68^{\circ}\). Thus, the lighthouse, the ship, and the plane form a straight line.
02

Calculate the Distance of the Ship from the Lighthouse

In the triangle formed by the lighthouse, ground, and the ship, which we know is a right triangle, we can use tangent of the angle, which is the ratio of the opposite side to the adjacent side. The formula is \(\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}\). Applying the formula, we get \(\tan(35^{\circ}) = \frac{250\text{ft}}{\text{distance}}\). Solving this for the distance, we find that the distance of the ship from the lighthouse is approximately \(366\text{ft}\).
03

Calculate the Plane's Height Above the Water

In the triangle formed by the plane, the lighthouse, and the point directly underneath the plane on the ground, we again use the tangent of the angle. Thus, we have \(\tan(68^{\circ}) = \frac{\text{height}}{366\text{ft}}\). Solving this for the height, we find that the plane's height above the water is roughly \(793\text{ft}\). Adding the height of the lighthouse to this, we get that the total height of the plane from the water is \(793\text{ft} + 250\text{ft} = 1043\text{ft}\).

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!

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

Use the Pythagorean Theorem to solve Exercises 39-46. Use your calculator to find square roots, rounding, if necessary, to the nearest tenth. A rectangular garden bed measures 5 feet by 12 feet. A water faucet is located at one corner of the garden bed. A hose will be connected to the water faucet. The hose must be long enough to reach the opposite corner of the garden bed when stretched straight. Find the required length of hose.

What does it mean if a graph is traversable?

Research non-Euclidean geometry and plan a seminar based on your group's research. Each group member should research one of the following five areas: a. Present an overview of the history of the people who developed non- Euclidean geometry. Who first used the term and why did he never publish his work? b. Present an overview of the connection between Saccheri quadrilaterals and non-Euclidean geometry. Describe the work of Girolamo Saccheri. c. Describe how Albert Einstein applied the ideas of Gauss and Riemann. Discuss the notion of curved space and a fourth dimension. d. Present examples of the work of M. C. Escher that provide ways of visualizing hyperbolic and elliptic geometry. e. Describe how non-Euclidean geometry changed the direction of subsequent research in mathematics. After all research has been completed, the group should plan the order in which each group member will speak. Each person should plan on taking about five minutes for his or her portion of the presentation.

The Washington Monument is 555 feet high. If you stand one quarter of a mile, or 1320 feet, from the base of the monument and look to the top, find the angle of elevation to the nearest degree.

Use the Pythagorean Theorem to solve Exercises 39-46. Use your calculator to find square roots, rounding, if necessary, to the nearest tenth. The base of a 20-foot ladder is 15 feet from the house. How far up the house does the ladder reach?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

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.