Problem 2. 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 = 2.62 miles and b = 3.91 miles. Round your answer to two decimal places.) B a 40.3° b
Problem 2. 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 = 2.62 miles and b = 3.91 miles. Round your answer to two decimal places.) B a 40.3° b
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Problem 2
To find the distance across a small lake, a surveyor has taken the measurements shown in the diagram. Find the distance across the lake using this information. (Assume points A and B are exactly along the shoreline, and that \( a = 2.62 \) miles and \( b = 3.91 \) miles. Round your answer to two decimal places.)
#### Diagram Description
The diagram accompanying the problem is a geometric representation with the following elements:
- A labeled shoreline and a small lake indicated with a blue shape.
- Point \( A \) and point \( B \) are situated exactly along the shoreline.
- Point \(C\) is another point on land, forming a triangle \( \Delta ABC \).
- The lengths of the segments \( AC \) and \( BC \) are represented as \( b \) and \( a \) respectively.
- \( a = 2.62 \) miles (distance \( BC \))
- \( b = 3.91 \) miles (distance \( AC \))
- Angle \( \angle ACB \) is given as \( 40.3^\circ \).
#### How to Solve
To find the distance across the lake, which is the direct distance between points \( A \) and \( B \), use the Law of Cosines:
\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\]
Where:
- \( c \) is the distance across the lake,
- \( a = 2.62 \) miles and \( b = 3.91 \) miles,
- \( C = 40.3^\circ \).
### Detailed Steps
1. Calculate \( c^2 \):
\[
c^2 = (2.62)^2 + (3.91)^2 - 2 \cdot 2.62 \cdot 3.91 \cdot \cos(40.3^\circ)
\]
2. Evaluate the trigonometric component:
\[
\cos(40.3^\circ) \approx 0.764
\]
3. Substitute the values:
\[
c^2 = 6.8644 + 15.2881 - 2 \cdot 2.62 \cdot 3.91 \cdot 0.764
\]
4. Simplify the expression:
\[](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc87785b0-7b08-4187-be5d-f53b11d0a1a0%2Faed3ac8f-5787-449e-bb3f-71aad1628a2d%2Fkdlbfze_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 2
To find the distance across a small lake, a surveyor has taken the measurements shown in the diagram. Find the distance across the lake using this information. (Assume points A and B are exactly along the shoreline, and that \( a = 2.62 \) miles and \( b = 3.91 \) miles. Round your answer to two decimal places.)
#### Diagram Description
The diagram accompanying the problem is a geometric representation with the following elements:
- A labeled shoreline and a small lake indicated with a blue shape.
- Point \( A \) and point \( B \) are situated exactly along the shoreline.
- Point \(C\) is another point on land, forming a triangle \( \Delta ABC \).
- The lengths of the segments \( AC \) and \( BC \) are represented as \( b \) and \( a \) respectively.
- \( a = 2.62 \) miles (distance \( BC \))
- \( b = 3.91 \) miles (distance \( AC \))
- Angle \( \angle ACB \) is given as \( 40.3^\circ \).
#### How to Solve
To find the distance across the lake, which is the direct distance between points \( A \) and \( B \), use the Law of Cosines:
\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\]
Where:
- \( c \) is the distance across the lake,
- \( a = 2.62 \) miles and \( b = 3.91 \) miles,
- \( C = 40.3^\circ \).
### Detailed Steps
1. Calculate \( c^2 \):
\[
c^2 = (2.62)^2 + (3.91)^2 - 2 \cdot 2.62 \cdot 3.91 \cdot \cos(40.3^\circ)
\]
2. Evaluate the trigonometric component:
\[
\cos(40.3^\circ) \approx 0.764
\]
3. Substitute the values:
\[
c^2 = 6.8644 + 15.2881 - 2 \cdot 2.62 \cdot 3.91 \cdot 0.764
\]
4. Simplify the expression:
\[
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