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Sagot :
To solve this problem, we will use trigonometry, specifically the tangent function. Here's a detailed, step-by-step solution:
1. Understand the Problem:
- We have the angle of depression from the ship to the wreck on the ocean floor: [tex]\(13.2^\circ\)[/tex].
- The depth directly below the ship to the ocean floor (i.e., the vertical distance) is [tex]\(75\)[/tex] meters.
- We need to find the distance along the ocean floor from the point directly below the ship to the wreck, rounding to the nearest tenth.
2. Visualize the Scenario:
- Imagine a right-angled triangle where:
- The depth below the ship is one leg of the triangle ([tex]\(75\)[/tex] meters, the opposite side relative to the angle of depression).
- The distance from the point directly below the ship to the wreck on the ocean floor is the adjacent side.
- The angle of depression ([tex]\(13.2^\circ\)[/tex]) is the angle between the horizontal line at the ship and the line of sight to the wreck.
3. Apply Trigonometry:
- The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side:
[tex]\[ \tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
In our case:
[tex]\[ \tan(13.2^\circ) = \frac{75}{\text{Distance to the wreck}} \][/tex]
4. Solve for the Distance to the Wreck:
- Let [tex]\( D \)[/tex] be the distance from the point directly below the ship to the wreck on the ocean floor:
[tex]\[ \tan(13.2^\circ) = \frac{75}{D} \][/tex]
- Rearrange the equation to solve for [tex]\( D \)[/tex]:
[tex]\[ D = \frac{75}{\tan(13.2^\circ)} \][/tex]
5. Calculate Using the Given Angle:
- Compute [tex]\( \tan(13.2^\circ) \)[/tex]:
[tex]\( \tan(13.2^\circ) \approx 0.2303834612632515 \)[/tex]
- Substitute the value into the equation:
[tex]\[ D = \frac{75}{0.2303834612632515} \approx 319.76413175085906 \text{ meters} \][/tex]
6. Round to the Nearest Tenth:
- To the nearest tenth, the distance [tex]\( D \)[/tex] is approximately [tex]\( 319.8 \)[/tex] meters.
7. Conclusion:
- The rounded distance from the point on the ocean floor to the wreck is approximately [tex]\( 319.8 \)[/tex] meters.
Therefore, the answer is:
[tex]\[ \text{C. } 319.8 \text{ meters} \][/tex]
1. Understand the Problem:
- We have the angle of depression from the ship to the wreck on the ocean floor: [tex]\(13.2^\circ\)[/tex].
- The depth directly below the ship to the ocean floor (i.e., the vertical distance) is [tex]\(75\)[/tex] meters.
- We need to find the distance along the ocean floor from the point directly below the ship to the wreck, rounding to the nearest tenth.
2. Visualize the Scenario:
- Imagine a right-angled triangle where:
- The depth below the ship is one leg of the triangle ([tex]\(75\)[/tex] meters, the opposite side relative to the angle of depression).
- The distance from the point directly below the ship to the wreck on the ocean floor is the adjacent side.
- The angle of depression ([tex]\(13.2^\circ\)[/tex]) is the angle between the horizontal line at the ship and the line of sight to the wreck.
3. Apply Trigonometry:
- The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side:
[tex]\[ \tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
In our case:
[tex]\[ \tan(13.2^\circ) = \frac{75}{\text{Distance to the wreck}} \][/tex]
4. Solve for the Distance to the Wreck:
- Let [tex]\( D \)[/tex] be the distance from the point directly below the ship to the wreck on the ocean floor:
[tex]\[ \tan(13.2^\circ) = \frac{75}{D} \][/tex]
- Rearrange the equation to solve for [tex]\( D \)[/tex]:
[tex]\[ D = \frac{75}{\tan(13.2^\circ)} \][/tex]
5. Calculate Using the Given Angle:
- Compute [tex]\( \tan(13.2^\circ) \)[/tex]:
[tex]\( \tan(13.2^\circ) \approx 0.2303834612632515 \)[/tex]
- Substitute the value into the equation:
[tex]\[ D = \frac{75}{0.2303834612632515} \approx 319.76413175085906 \text{ meters} \][/tex]
6. Round to the Nearest Tenth:
- To the nearest tenth, the distance [tex]\( D \)[/tex] is approximately [tex]\( 319.8 \)[/tex] meters.
7. Conclusion:
- The rounded distance from the point on the ocean floor to the wreck is approximately [tex]\( 319.8 \)[/tex] meters.
Therefore, the answer is:
[tex]\[ \text{C. } 319.8 \text{ meters} \][/tex]
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