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A boat is heading towards a lighthouse, whose beacon-light is 108 feet above the water. From point A, the boat’s crew measures the angle of elevation to the beacon, 8∘, before they draw closer. They measure the angle of elevation a second time from point B at some later time to be 16∘. Find the distance from point A to point B. Round your answer to the nearest foot if necessary.

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Using the slope concept, it is found that the distance from point A to point B is of 392 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:

  • 108 feet above the water, hence a vertical change is of 108.
  • The horizontal position is [tex]x_A[/tex], which we want to find.
  • The angle of depression is of 8º.

Then:

[tex]\tan{8^{\circ}} = \frac{108}{x_A}[/tex]

[tex]x_A = \frac{108}{\tan{8^{\circ}}}[/tex]

[tex]x_A = 768.46[/tex]

At point B:

  • 108 feet above the water, hence a vertical change is of 108.
  • The horizontal position is [tex]x_B[/tex], which we want to find.
  • The angle of depression is of 16º.

Then:

[tex]\tan{16^{\circ}} = \frac{108}{x_B}[/tex]

[tex]x_B = \frac{108}{\tan{16^{\circ}}}[/tex]

[tex]x_B = 376.64[/tex]

The distance is:

[tex]d = x_A - x_B = 768.46 - 376.64 \approx 392[/tex]

The distance from point A to point B is of 392 feet.

You can learn more about the slope concept at https://brainly.com/question/18090623

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