A boat is heading towards a lighthouse, whose beacon-light is 113 feet above the water. From point AA, the boat’s crew measures the angle of elevation to the beacon, 10^{\circ} ∘ , before they draw closer. They measure the angle of elevation a second time from point BB at some later time to be 21^{\circ} ∘ . Find the distance from point AA to point BB. Round your answer to the nearest foot if necessary.

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MrRoyal MrRoyal · Answer 1 · 2022-03-04T11:03:23+01:00

The distance between points A and B and the boat is an illustration of elevation and distance

The distance from point A to point B is 346 feet

The height of the lighthouse is given as:

h = 113

The distance between point A and the base of the lighthouse is calculated using the following tangent ratio

[tex]\tan(10) = \frac{113}{A}[/tex]

Make A, the subject

[tex]A = \frac{113}{\tan(10)}[/tex]

[tex]A = 640.85[/tex]

The distance between point B and the base of the lighthouse is calculated using the following tangent ratio

[tex]\tan(21) = \frac{113}{B}[/tex]

Make B, the subject

[tex]B= \frac{113}{\tan(21)}[/tex]

[tex]B= 294.38[/tex]

The distance AB, is then calculated as:

[tex]AB = A - B[/tex]

[tex]AB = 640.85 - 294.38[/tex]

[tex]AB = 346.47[/tex]

Approximate

[tex]AB = 346[/tex]

Hence, the distance from point A to point B is 346 feet

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