Question

From the obserdation deck of a skyscraper, Nakeisha measures a 51° angle of
depression to a ship in the harbor below. If the observation deck is 1166 feet high,
what is the horizontal distance from the base of the skyscraper out to the ship? Round
your answer to the nearest tenth of a foot if necessary.

Answer 1

The horizontal distance from the base of the skyscraper out to the ship is 944.1 feet approximately.

What is angle of depression?

You look straight parallel to ground. But when you have to watch something down, then you take your sight down by moving your head up. The angle from horizontal to the point where you stopped your head is called angle of depression.

See the figure where angle DAC is the angle when Nakeisha saw the boat from the top of the skyscrapper. It  is measured from the horizontal line AD.

For this case, referring to the figure attached below, we see that:

$m\angle ACB = m\angle DAC = 51^\circ$ (because of them being alternate interior angles). (the small 'm' ahead of angles show that we're taking about measurement of angles).

Thus, using the tangent ratio from the perspective of $\angle ACB$, we get:

$\tan(\angle ACB) = \dfrac{|AB|}{|CB|}\\\\\\\tan(51^\circ) = \dfrac{1166}{x}$

where |AB| denotes the length of the line segment AB.

From calculator, we get $\tan(51^\circ) \approx 1.235$

Thus, we get:

$\tan(51^\circ) = \dfrac{1166}{x}\\\\1.235 \approx \dfrac{1166}{x}\\\\x \approx \dfrac{1166}{1.235} \approx 944.129 \approx 944.1 \: \rm ft$

Thus, the horizontal distance from the base of the skyscraper out to the ship is 944.1 feet approximately.

Learn more about tangent ratio here:

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