19. A ship's sonar finds that the angle of depression to a wreck on the
bottom of the ocean is 13.2°. If a
point on the ocean floor is 75
meters directly below the ship
, how many meters is it from that
point on the ocean floor
to the wreck? Round to the nearest tenth.
C. 319.8 meters
A. 328.4 meters
B. 77.0 meters
D. 17.6 meters



Answer :

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]