**Problem Statement:**A boat is sailing due east parallel to the shoreline at a speed of 11 miles per hour. At a given time, the bearing to the lighthouse is S 70° E, and 15 minutes later the bearing is S 63° E (see figure). The lighthouse is located at the shoreline. What is the distance from the boat to the shoreline? (Round your answer to one decimal place.)**Diagram Explanation:**- The figure in the image shows two positions of a boat while sailing east, parallel to the shoreline: - The boat’s initial position marks an angle of 70° to the lighthouse, with the lighthouse situated at the shoreline. - After 15 minutes, the boat moves further to the east and marks an angle of 63° to the lighthouse.- An inset compass indicates directions for navigational bearings, with the eastward direction clearly labeled.**Calculation Details:**- Let $ d $ be the distance from the boat to the shoreline.- Since the speed of the boat is 11 miles per hour: - In 15 minutes (which is $\frac{1}{4}$ of an hour), the boat travels $ \frac{11}{4} = 2.75 $ miles.- Using trigonometric relationships, we can employ the following steps to find $ d $:**Figure and Angle Analysis:** 1. **Form a Right Triangle:** - Let point A represent the boat's initial position, B represent the boat's position after 15 minutes, and C represent the lighthouse at the shoreline. - The distance $ AB = 2.75 $ miles. - Angle $ \angle BAC = 70° $ - Angle $ \angle ABC = 63° $**Solution Process:** 1. Determine angle $ \angle ACB $, the third angle in triangle ACB: \[ \angle ACB = 180° - 70° - 63° = 47° \] 2. Use the Law of Sines to find side `d`: The Law of Sines states: \[ \frac{AB}{\sin(\angle ACB)} = \frac{d}{\sin(\angle ABC)} \] Plugging in the values: \[ \frac{2.75}{\sin(47

Answered: Image /qna-images/answer/c4840dc3-5561-4704-8010-2dd81e683162.jpg

A boat is sailing due east parallel to the shoreline at a speed of 11 miles per hour. At a given time, the bearing to the lighthouse is S 70° E, and 15 minutes later the bearing is S 63° E (see figure). The lighthouse is located at the shoreline. What is the distance from the boat to the shoreline? (Round your answer to one decimal place.) mi 63° W 70°

A boat is sailing due east parallel to the shoreline at a speed of 11 miles per hour. At a given time, the bearing to the lighthouse is S 70° E, and 15 minutes later the bearing is S 63° E (see figure). The lighthouse is located at the shoreline. What is the distance from the boat to the shoreline? (Round your answer to one decimal place.) mi 63° W 70°

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

**Problem Statement:**

A boat is sailing due east parallel to the shoreline at a speed of 11 miles per hour. At a given time, the bearing to the lighthouse is S 70° E, and 15 minutes later the bearing is S 63° E (see figure). The lighthouse is located at the shoreline. What is the distance from the boat to the shoreline? (Round your answer to one decimal place.)

**Diagram Explanation:**

- The figure in the image shows two positions of a boat while sailing east, parallel to the shoreline:
- The boat’s initial position marks an angle of 70° to the lighthouse, with the lighthouse situated at the shoreline.
- After 15 minutes, the boat moves further to the east and marks an angle of 63° to the lighthouse.
- An inset compass indicates directions for navigational bearings, with the eastward direction clearly labeled.

**Calculation Details:**

- Let $ d $ be the distance from the boat to the shoreline.
- Since the speed of the boat is 11 miles per hour:
- In 15 minutes (which is $\frac{1}{4}$ of an hour), the boat travels $ \frac{11}{4} = 2.75 $ miles.
- Using trigonometric relationships, we can employ the following steps to find $ d $:

**Figure and Angle Analysis:**

1. **Form a Right Triangle:**
- Let point A represent the boat's initial position, B represent the boat's position after 15 minutes, and C represent the lighthouse at the shoreline.
- The distance $ AB = 2.75 $ miles.
- Angle $ \angle BAC = 70° $
- Angle $ \angle ABC = 63° $

**Solution Process:**

1. Determine angle $ \angle ACB $, the third angle in triangle ACB:
\[ \angle ACB = 180° - 70° - 63° = 47° \]

2. Use the Law of Sines to find side `d`:
The Law of Sines states:
\[
\frac{AB}{\sin(\angle ACB)} = \frac{d}{\sin(\angle ABC)}
\]
Plugging in the values:
\[
\frac{2.75}{\sin(47

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