Use the image to answer the question.
A ship’s captain sights the top of a 200 foot cliff at a
17
°
angle of elevation. How far is the ship from the base of the cliff? Round to the nearest foot.
A right triangle is shown, formed by a boat near an acute angle labeled 17 degrees, side x along the bottom leg, and the other leg labeled 200ft to represent the cliff.
A.
58 feet
B.
191 feet
C.
654 feet
D.
684 feet
B. 191 feet
To find the distance from the ship to the base of the cliff, we need to find the length of the side opposite the angle of elevation. This side represents the height of the cliff.
Using trigonometry, we can use the tangent function to find the length of this side.
The tangent of an angle is equal to the opposite side divided by the adjacent side.
In this case, the opposite side is the height of the cliff (200 ft) and the adjacent side is the distance from the ship to the base of the cliff (x ft).
So, we have:
tan(17°) = opposite/adjacent
tan(17°) = 200/x
To solve for x, we can rearrange the equation:
x = 200 / tan(17°)
Using a calculator, we find:
x ≈ 654.2
Rounding to the nearest foot, the ship is about 654 feet from the base of the cliff.
Therefore, the answer is C. 654 feet.