From the observation deck of a skyscraper, Valeria measures a 67degrees ∘ angle of depression to a ship in the harbor below. If the observation deck is 945 feet high, what is the horizontal distance from the base of the skyscraper out to the ship? Round your answer to the nearest hundredth of a foot if necessary.
Question
From the observation deck of a skyscraper, Valeria measures a 67degrees ∘ angle of depression to a ship in the harbor below. If the observation deck is 945 feet high, what is the horizontal distance from the base of the skyscraper out to the ship? Round your answer to the nearest hundredth of a foot if necessary.
Solution
To solve this problem, we can use the tangent of the angle of depression, which is the ratio of the opposite side (height of the skyscraper) to the adjacent side (horizontal distance we want to find).
The formula for tangent is tan(θ) = opposite/adjacent.
First, we need to note that the angle of depression is measured from a horizontal line of sight. If we draw this scenario, we have a right triangle where the height of the skyscraper is the side opposite to the angle of elevation (which is complementary to the angle of depression and hence is also 67 degrees).
So, we can set up the equation as follows:
tan(67) = 945 / x
To solve for x (the horizontal distance from the base of the skyscraper to the ship), we rearrange the equation:
x = 945 / tan(67)
Now, we just plug in the values and solve:
x = 945 / tan(67) ≈ 355.29 feet
So, the horizontal distance from the base of the skyscraper out to the ship is approximately 355.29 feet.
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