A boat has passed directly over a submarine as pictured below. The sub is running at a depth of 588 ft, and the angle of depression from the boat to the sub is 35 degrees. Find the distance, x, from the boat to the submarine.
To find the distance, x, from the boat to the submarine, we can use trigonometry. Specifically, we can use the tangent function as it relates the angle of depression and the opposite side of a right triangle.
Let's assume that the distance, x, is the adjacent side and the depth of the submarine, 588 ft, is the opposite side of the right triangle formed. The angle of depression, 35 degrees, is the angle between the hypotenuse and the adjacent side.
Using the tangent function, we have:
tan(35 degrees) = opposite / adjacent
tan(35 degrees) = 588 ft / x
To solve for x, we can rearrange the equation:
x = 588 ft / tan(35 degrees)
Using a calculator, we can find the value of tan(35 degrees) ≈ 0.7002.
Therefore, substituting this value into the equation:
x = 588 ft / 0.7002
Calculating this, we find that x ≈ 839.4 ft.
So, the distance from the boat to the submarine, x, is approximately 839.4 ft.
To find the distance, x, from the boat to the submarine, we can use trigonometry.
In this case, the angle of depression is the angle between the horizontal line (i.e., the water's surface) and the line of sight from the boat to the submarine. This angle is 35 degrees.
We are given that the submarine is running at a depth of 588 ft, which is the height of the right-angled triangle formed by the boat, the submarine, and the water's surface.
Let's label the distance from the boat to the submarine as x.
Since the angle of depression is 35 degrees, we can use the tangent function:
tan(35 degrees) = opposite/adjacent
In this case, the opposite side is the depth of the submarine (588 ft) and the adjacent side is the distance from the boat to the submarine (x).
Therefore:
tan(35 degrees) = 588 ft / x
To find x, we can rearrange the equation:
x = 588 ft / tan(35 degrees)
Using a scientific calculator, we can calculate x:
x ≈ 588 ft / tan(35 degrees)
x ≈ 588 ft / 0.7002
x ≈ 839.25 ft
Therefore, the distance from the boat to the submarine, x, is approximately 839.25 feet.