**Problem Statement:**To find the distance across a small lake, a surveyor has taken the measurements shown. Find the distance across the lake using this information. (Assume points \( A \) and \( B \) are exactly along the shoreline, and that \( a = 2.72 \) miles and \( b = 3.91 \) miles. Round your answer to two decimal places.)**Diagram Explanation:**The diagram shows a triangle \( \triangle ABC \) with:- Side \( a \) opposite angle \( ACB \) measuring 2.72 miles.- Side \( b \) opposite angle \( BAC \) measuring 3.91 miles.- Angle \( ACB \) measuring 40.3°.The lake is depicted as lying between points \( A \) and \( B \) along the shoreline, with the triangle crossing the lake. The task is to find the length of the side \( AB \).(Fill in the calculated distance in miles in the provided box.) **Problem Statement:**A water tower 30 meters tall is located at the top of a hill. From a distance of \( D = 145 \) meters down the hill, it is observed that the angle formed between the top and base of the tower is \( 8^\circ \). Find the angle of inclination of the hill. (Round your answer to one decimal place.)**Image Explanation:**The diagram shows a hill with a water tower positioned at the top. The water tower is 30 meters tall. From a point 145 meters down the hill, the line of sight forms an \( 8^\circ \) angle with the horizontal line from the base to the top of the tower. The hill's slope is represented by a hypotenuse extending from the observation point to the top of the tower.**Instructions for Solving:**1. Identify the relevant right triangle formed by the height of the tower, the distance \( D \), and the hypotenuse along the hill.2. Use trigonometric principles to calculate the angle of inclination of the hill, taking note of the given \( 8^\circ \) angle to calculate the desired angle.3. Use trigonometric relationships such as tangent, sine, or cosine as required for solving the problem.4. Round the result to one decimal place.

Since you have asked multiple question, we will solve the first question for you. If youwant any…

To find the distance across a small lake, a surveyor has taken the measurements shown. Find the distance across the lake using this information. (Assume points A and B are exactly along the shoreline, and that a = 2.72 miles and b = 3.91 miles. Round your answer to two decimal places.) mi B a 40.3° b.

To find the distance across a small lake, a surveyor has taken the measurements shown. Find the distance across the lake using this information. (Assume points A and B are exactly along the shoreline, and that a = 2.72 miles and b = 3.91 miles. Round your answer to two decimal places.) mi B a 40.3° b.

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:**

To find the distance across a small lake, a surveyor has taken the measurements shown. Find the distance across the lake using this information. (Assume points \( A \) and \( B \) are exactly along the shoreline, and that \( a = 2.72 \) miles and \( b = 3.91 \) miles. Round your answer to two decimal places.)

**Diagram Explanation:**

The diagram shows a triangle \( \triangle ABC \) with:

- Side \( a \) opposite angle \( ACB \) measuring 2.72 miles.
- Side \( b \) opposite angle \( BAC \) measuring 3.91 miles.
- Angle \( ACB \) measuring 40.3°.

The lake is depicted as lying between points \( A \) and \( B \) along the shoreline, with the triangle crossing the lake. The task is to find the length of the side \( AB \).

(Fill in the calculated distance in miles in the provided box.)

**Problem Statement:**

A water tower 30 meters tall is located at the top of a hill. From a distance of \( D = 145 \) meters down the hill, it is observed that the angle formed between the top and base of the tower is \( 8^\circ \). Find the angle of inclination of the hill. (Round your answer to one decimal place.)

**Image Explanation:**

The diagram shows a hill with a water tower positioned at the top. The water tower is 30 meters tall. From a point 145 meters down the hill, the line of sight forms an \( 8^\circ \) angle with the horizontal line from the base to the top of the tower. The hill's slope is represented by a hypotenuse extending from the observation point to the top of the tower.

**Instructions for Solving:**

1. Identify the relevant right triangle formed by the height of the tower, the distance \( D \), and the hypotenuse along the hill.
2. Use trigonometric principles to calculate the angle of inclination of the hill, taking note of the given \( 8^\circ \) angle to calculate the desired angle.
3. Use trigonometric relationships such as tangent, sine, or cosine as required for solving the problem.
4. Round the result to one decimal place.

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