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A scuba diver located 20 feet below the surface of the water spots a ship wreck at a 70 degree angle of depression. After descending to a point 45 ft above the in ocean floor, the diver sees the shipwreck at a 57 degree angle of depression. Determine the depth of the shipwreck



Answer :

Answer:

The depth of the shipwreck is 100.29 feet.

Explanation:

We can determine the depth of the shipwreck by looking at the attached picture:

  • Point A is where the diver is 20 feet below the surface.
  • Point B is where the diver is 45 feet above the ocean floor.
  • Point C is where the shipwreck is located.

For ΔACD:

[tex]\displaystyle tan(70^o)=\frac{AC}{CD}[/tex]

[tex]\displaystyle tan(70^o)=\frac{45+y}{x}[/tex]

[tex]\displaystyle x=\frac{45+y}{tan(70^o)}[/tex] ... [1]

For ΔBCD:

[tex]\displaystyle tan(57^o)=\frac{BC}{CD}[/tex]

[tex]\displaystyle tan(57^o)=\frac{45}{x}[/tex]

[tex]\displaystyle x=\frac{45}{tan(57^o)}[/tex] ... [2]

Since [tex]x[/tex] in [1] equals to [tex]x[/tex] in [2], then:

[tex]\displaystyle \frac{45+y}{tan(70^o)}=\frac{45}{tan(57^o)}[/tex]

[tex]\displaystyle 45+y=\frac{tan(70^o)(45)}{tan(57^o)}[/tex]

[tex]\displaystyle y=\frac{tan(70^o)(45)}{tan(57^o)}-45[/tex]

[tex]y=35.29\ ft[/tex]

The depth of the shipwreck:

[tex]depth=20+y+45[/tex]

         [tex]=20+35.29+45[/tex]

         [tex]=\bf 100.29\ ft[/tex]

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