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: \(\(13.2^\circ\)\).
- The depth directly below the ship to the ocean floor (i.e., the vertical distance) is \(\(75\)\) 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 (\(\(75\)\) 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 (\(\(13.2^\circ\)\)) 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:
\(\[
\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}
\]\)
In our case:
\(\[
\tan(13.2^\circ) = \frac{75}{\text{Distance to the wreck}}
\]\)
4. Solve for the Distance to the Wreck:
- Let \(\( D \)\) be the distance from the point directly below the ship to the wreck on the ocean floor:
\(\[
\tan(13.2^\circ) = \frac{75}{D}
\]\)
- Rearrange the equation to solve for \(\( D \)\):
\(\[
D = \frac{75}{\tan(13.2^\circ)}
\]\)
5. Calculate Using the Given Angle:
- Compute \(\( \tan(13.2^\circ) \)\):
\(\( \tan(13.2^\circ) \approx 0.2303834612632515 \)\)
- Substitute the value into the equation:
\(\[
D = \frac{75}{0.2303834612632515} \approx 319.76413175085906 \text{ meters}
\]\)
6. Round to the Nearest Tenth:
- To the nearest tenth, the distance \(\( D \)\) is approximately \(\( 319.8 \)\) meters.
7. Conclusion:
- The rounded distance from the point on the ocean floor to the wreck is approximately \(\( 319.8 \)\) meters.
Therefore, the answer is:
\(\[ \text{C. } 319.8 \text{ meters} \]\)
