A forest ranger in a 120ft observation tower sees a fire. The angle of depression to the fire is 3.5. What is the horizontal distance between the tower and the fire
horizontal distance --- x
I see it as:
tan 3.5° = 120/x
x = 120/tan3.5 = appr 1962 ft
To find the horizontal distance between the tower and the fire, we can use trigonometric ratios. In this case, we'll use the tangent ratio.
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 scenario, the opposite side is the height of the tower (120 ft) and the adjacent side is the horizontal distance we're trying to find.
We know the angle of depression to the fire is 3.5 degrees. Let's represent the horizontal distance as "x".
Using the tangent ratio:
tan(angle) = opposite/adjacent
tan(3.5) = 120/x
We can now solve for x by rearranging the equation:
x = 120 / tan(3.5)
Using a calculator, we can find the value of "x":
x ≈ 2055.328
Thus, the horizontal distance between the tower and the fire is approximately 2055.328 ft.
To find the horizontal distance between the observation tower and the fire, we can use the tangent function.
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, the angle of depression of 3.5 degrees gives us the opposite side (the height of the observation tower) and we need to find the adjacent side (the horizontal distance).
Let's denote the horizontal distance as "d", and the height of the observation tower as "h".
Using tangent, we can set up the following equation:
tan(3.5) = h / d
To solve for "d", we need to isolate it on one side of the equation. We can do this by rearranging the equation:
d = h / tan(3.5)
Now, we can substitute the given information into the equation:
h = 120 ft
angle of depression = 3.5 degrees
Plugging these values into the equation, we get:
d = 120 ft / tan(3.5)
Using a scientific calculator, we can calculate the tangent of 3.5 degrees:
tan(3.5) ≈ 0.0610865
Now, substitute this value into the equation:
d = 120 ft / 0.0610865
Evaluating this expression gives us:
d ≈ 1963.37 ft
Therefore, the horizontal distance between the observation tower and the fire is approximately 1963.37 feet.