A 2.05m man, standing in front of a street light 3.08m high, casts a 1.5m shadow. What is the angle of elevation from the ground to the source of light?
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To find the angle of elevation from the ground to the source of light, we can use basic trigonometry.
Let's denote the height of the man as h1, the height of the street light as h2, and the length of the shadow as s.
We have the following information:
h1 = 2.05m
h2 = 3.08m
s = 1.5m
To find the angle of elevation, we need to find the ratio of the opposite side (h1) to the adjacent side (s), which is tan(θ) = h1 / s.
Thus, θ = tan^(-1) (h1 / s).
Substituting the given values into the formula, we get:
θ = tan^(-1) (2.05 / 1.5)
Using a calculator, we find:
θ ≈ 53.13 degrees
Therefore, the angle of elevation from the ground to the source of light is approximately 53.13 degrees.
To find the angle of elevation, we can use similar triangles. Let's call the height of the man H and the height of the street light h. We are given that H = 2.05 m, h = 3.08 m, and the length of the shadow is 1.5 m.
First, we need to find the length of the man's shadow on the ground. Since the triangles formed by the man, his shadow, and the street light are similar, we can set up the following proportion:
(H + h) / h = (length of shadow + length of man) / length of shadow
Plugging in the known values, we get:
(2.05 + 3.08) / 3.08 = (1.5 + length of man) / 1.5
5.13 / 3.08 = (1.5 + length of man) / 1.5
Cross-multiplying, we get:
5.13 * 1.5 = 3.08 * (1.5 + length of man)
7.695 = 4.62 + 3.08 * length of man
Subtracting 4.62 from both sides, we have:
7.695 - 4.62 = 3.08 * length of man
3.075 = 3.08 * length of man
Now, we can solve for the length of the man:
length of man = 3.075 / 3.08 = 0.99675 meters
Now that we have the length of the man, we can find the angle of elevation. The angle of elevation is the angle formed between the line of sight from the man's feet to the source of light, and the horizontal ground.
We can use trigonometry to find this angle. The tangent of the angle of elevation is equal to the height of the man divided by the length of the shadow.
tan(angle of elevation) = height of man / length of shadow
Plugging in the values, we get:
tan(angle of elevation) = 2.05 / 1.5
Solving for the angle of elevation, we take the inverse tangent (arctan) of both sides:
angle of elevation = arctan(2.05 / 1.5) ≈ 53.13 degrees
Therefore, the angle of elevation from the ground to the source of light is approximately 53.13 degrees.
review your basic trig functions, and you will note that
tanθ = 2.05/1.5
The height of the light does not really matter. It will just help to determine how far away from the lamp the man is standing.