Nicole shines a light from a window of a lighthouse on a cliff 250 feet above the water level. Nick 10 feet above the water level in a ship off shore, finds that the angle of elevation of the light is 3°. Find the length of the line of sight (light beam) from the ship to Nicole.

If we draw the triangle, the opposite side to 3° angle would be "10" less than total height of 250 because Nick is 10 feet above water level, so that side will be:

250 - 10 = 240

The hypotenuse of the triangle is the length of line of sight. We can call this "x".

So, using trigonometric ratio of sine (opposite/hypotenuse), we can write:

[tex]Sin(3)=\frac{240}{x}[/tex]

Now, we cross multiply and solve for x, line of sight length:

[tex]Sin(3)=\frac{240}{x}\\x=\frac{240}{Sin(3)}\\x=4585.8[/tex]

facundo3141592 facundo3141592 · Answer 2 · 2021-11-29T10:39:40+01:00

We want to find the length of the light beam from the ship to Nicole.

We will see that the solution is 4,585.76 ft.

We can think of this situation as in a right triangle. The adjacent cathetus is the distance between the lighthouse and the ship. The opposite cathetus is the difference in height between Nicole's position and Nick´s position, it is equal to:

250ft - 10ft = 240ft

And the elevation angle is equal to 3°. So we know the angle and the opposite cathetus to this angle, and we want to find the hypotenuse, which is the length of the light beam.

Then we can use the relation:

Sin(θ) = (opposite cathetus)/(hypotenuse)

Solving it for the hypotenuse we get:

hypotenuse = (opposite cathetus)/sin(θ)

Replacing by the values that we know, we get:

hypotenuse = 240ft/sin(3°) = 4,585.76 ft

The length of the line of sight is 4,585.76 ft

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