A 200-foot cliff drops vertically into the ocean. The angle of elevation from a ship to the top of the cliff is 22.3 degrees. How far offshore, to the nearest foot, is the ship?
x/200 = cot 22.3°
To find the distance between the ship and the base of the cliff, which will give the distance offshore, we can use trigonometry.
Let's assume that the distance between the ship and the base of the cliff is x feet.
In a right triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.
In this case, we can use the tangent function to find x:
tan(22.3 degrees) = opposite/adjacent
tan(22.3 degrees) = 200 ft / x
To solve for x, we can rearrange the equation:
x = (200 ft) / tan(22.3 degrees)
Using a calculator, the approximate value of x is:
x ≈ (200 ft) / tan(22.3 degrees) = 600.72 ft
Therefore, the ship is approximately 600.72 feet offshore, to the nearest foot.
To find the distance between the ship and the cliff, we can use trigonometry. Specifically, we will use the tangent function because we have the angle of elevation and the opposite side (the height of the cliff). Here's how you can find the distance to the nearest foot:
1. Draw a right triangle. The vertical side represents the height of the cliff (200 feet), the horizontal side represents the distance from the ship to the cliff (unknown), and the angle of elevation is 22.3 degrees.
2. Identify the trigonometric function that relates the angle of elevation and the sides of the triangle. In this case, the tangent function relates the opposite side (height of the cliff) and the adjacent side (distance from the ship to the cliff).
3. Apply the tangent function: tan(angle) = opposite/adjacent. Substitute the known values into the equation: tan(22.3°) = 200/adjacent.
4. Solve for the adjacent side (distance from the ship to the cliff): adjacent = 200 / tan(22.3°).
5. Use a calculator to evaluate tan(22.3°) and divide 200 by the result.
* tan(22.3°) ≈ 0.407
* adjacent ≈ 200 / 0.407 ≈ 491.49 feet
6. Round the answer to the nearest foot: The ship is approximately 491 feet offshore.