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A boat is heading towards a lighthouse, where Camila is watching from a vertical
distance of 107 feet above the water. Camila measures an angle of depression to the
boat at point A to be 20°. At some later time, Camila takes another measurement
and finds the angle of depression to the boat (now at point B) to be 31°. Find the
distance from point A to point B. Round your answer to the nearest foot if
necessary.

Respuesta :

Using the slope concept, it is found that the distance from point A to point B is of 116 feet.

What is a slope?

  • The slope is given by the vertical change divided by the horizontal change.
  • It's also the tangent of the angle of depression.

At point A:

  • 107 feet above the water, hence a vertical change is of 107.
  • The horizontal position is [tex]x_A[/tex], which we want to find.
  • The angle of depression is of 20º.

Hence:

[tex]\tan{20^{\circ}} = \frac{107}{x_A}[/tex]

[tex]x_A\tan{20^{\circ}} = 107[/tex]

[tex]x_A = \frac{107}{\tan{20^{\circ}}}[/tex]

[tex]x_A = 294[/tex]

At point B:

  • 107 feet above the water, hence a vertical change is of 107.
  • The horizontal position is [tex]x_B[/tex], which we want to find.
  • The angle of depression is of 31º.

Hence:

[tex]\tan{31^{\circ}} = \frac{107}{x_B}[/tex]

[tex]x_B\tan{31^{\circ}} = 107[/tex]

[tex]x_B = \frac{107}{\tan{31^{\circ}}}[/tex]

[tex]x_B = 178[/tex]

The distance is:

[tex]D = |x_A - x_B| = |294 - 178| = 116[/tex]

The distance from point A to point B is of 116 feet.

You can learn more about the slope concept at https://brainly.com/question/26125945

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