A ship's sonar locates a treasure chest at a 12° angle of depression. A diver is lowered 40 meters to the ocean floor. How far does the diver need to swim along the ocean floor to get to the treasure chest? 12° 40 m 12°
A ship's sonar locates a treasure chest at a 12° angle of depression. A diver is lowered 40 meters to the ocean floor. How far does the diver need to swim along the ocean floor to get to the treasure chest? 12° 40 m 12°
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.
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Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.
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![### Locating a Treasure Chest Underwater
A ship's sonar locates a treasure chest at a 12° angle of depression. A diver is lowered 40 meters to the ocean floor. How far does the diver need to swim along the ocean floor to get to the treasure chest?
#### Diagram Explanation:
The diagram shows a right-angled triangle with the following components:
- The vertical side represents the depth at which the diver is lowered, which is 40 meters.
- The angle of depression from the ship to the treasure chest is 12°.
- The horizontal side, labeled as X, represents the distance the diver needs to swim along the ocean floor to reach the treasure chest.
- The hypotenuse is implied along the line of sight from the ship to the treasure chest.
To determine the distance \( X \) that the diver needs to swim, we can use trigonometric ratios. Specifically, the tangent function is useful here because it relates the angle of depression (or elevation), the opposite side (depth), and the adjacent side (distance to swim).
#### Calculation using Tangent Function:
\[
\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}
\]
For this problem:
\[
\tan(12°) = \frac{40 \text{ meters}}{X}
\]
Rearranging to solve for \( X \):
\[
X = \frac{40 \text{ meters}}{\tan(12°)}
\]
Using a calculator to find \( \tan(12°) \approx 0.2126 \):
\[
X = \frac{40 \text{ meters}}{0.2126} \approx 188.2 \text{ meters}
\]
So, the diver needs to swim approximately 188.2 meters along the ocean floor to reach the treasure chest.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F693046e0-0496-440e-a4a1-2d8d9c31eda7%2F9c6be8e8-a8f1-4cab-9631-ba07aa257b8f%2Fny1teln_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Locating a Treasure Chest Underwater
A ship's sonar locates a treasure chest at a 12° angle of depression. A diver is lowered 40 meters to the ocean floor. How far does the diver need to swim along the ocean floor to get to the treasure chest?
#### Diagram Explanation:
The diagram shows a right-angled triangle with the following components:
- The vertical side represents the depth at which the diver is lowered, which is 40 meters.
- The angle of depression from the ship to the treasure chest is 12°.
- The horizontal side, labeled as X, represents the distance the diver needs to swim along the ocean floor to reach the treasure chest.
- The hypotenuse is implied along the line of sight from the ship to the treasure chest.
To determine the distance \( X \) that the diver needs to swim, we can use trigonometric ratios. Specifically, the tangent function is useful here because it relates the angle of depression (or elevation), the opposite side (depth), and the adjacent side (distance to swim).
#### Calculation using Tangent Function:
\[
\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}
\]
For this problem:
\[
\tan(12°) = \frac{40 \text{ meters}}{X}
\]
Rearranging to solve for \( X \):
\[
X = \frac{40 \text{ meters}}{\tan(12°)}
\]
Using a calculator to find \( \tan(12°) \approx 0.2126 \):
\[
X = \frac{40 \text{ meters}}{0.2126} \approx 188.2 \text{ meters}
\]
So, the diver needs to swim approximately 188.2 meters along the ocean floor to reach the treasure chest.
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