From a boat on the lake, the angle of elevation to the top of a cliff is 28°13'. If the base of the cliff is 205 feet from the boat, how high is the cliff (to the nearest foot)?
Answers-
100 ft
133ft
103 ft
110 ft
h / 205 = tan(28º13')
To find the height of the cliff, we can use trigonometry. We have the angle of elevation and the distance from the boat to the base of the cliff. We can use the tangent function, which is defined as the opposite side divided by the adjacent side.
Let's calculate the height of the cliff step-by-step:
1. First, convert the angle of elevation to decimal form.
28°13' = 28 + (13/60) = 28.2167°
2. Next, apply the tangent function:
tan(angle) = opposite/adjacent
tan(28.2167°) = h/205
3. Rearrange the formula to solve for h (height):
h = 205 * tan(28.2167°)
4. Use a calculator to evaluate the expression:
h ≈ 101.9119 feet
The height of the cliff to the nearest foot is approximately 102 feet. Therefore, none of the given answer options matches the calculated height.
To solve this problem, we can use trigonometry. The angle of elevation and the distance from the boat to the base of the cliff form a right triangle. We can use the tangent function, which is the opposite over the adjacent, to find the height of the cliff.
First, we need to convert the angle of elevation from degrees and minutes to decimal form. The angle is given as 28°13'. To convert the minutes to decimal, we divide 13 by 60, which is 0.217.
So, the angle of elevation becomes 28° + 0.217 = 28.217°.
Next, we can use the tangent function. The formula is tangent(angle) = opposite / adjacent.
In this case, the opposite side is the height of the cliff, and the adjacent side is the distance from the boat to the base of the cliff, which is 205 feet.
So, we have the equation tan(28.217°) = height / 205.
Now, we can solve for the height by multiplying both sides of the equation by 205.
205 * tan(28.217°) = height.
Using a scientific calculator or an online trigonometric calculator, we can find that the value of tan(28.217°) is approximately 0.531. Therefore,
205 * 0.531 = height.
Height ≈ 108.755 feet.
Rounding to the nearest foot, the height of the cliff is 109 feet.
Therefore, the closest answer is 110 ft.