A 5.5 ft tall woman casts a 6.2 ft shadow. Find the angle of elevation of the sun.
tan (angle) = 5.5/6.2
tan^-1(5.5/6.2)
=41.6
^^^using a TI-84 PLUS^^^
To find the angle of elevation of the sun, we can use the concept of similar triangles.
Let's denote the height of the woman as h and the length of her shadow as s.
From the problem, we know that h = 5.5 ft and s = 6.2 ft.
First, we can set up the proportion between the corresponding sides of the similar triangles formed by the woman and her shadow:
h / s = H / S
Where H represents the height of the sun, and S represents the length of the shadow cast by the sun.
We want to find the angle of elevation, which is the angle between the vertical line (representing the height of the woman) and the line connecting the top of her head to the top of her shadow (representing the length of the shadow cast by the sun).
The tangent of this angle can be written as:
tan(angle) = h / s
Substituting the given values, we have:
tan(angle) = 5.5 ft / 6.2 ft
Next, we can calculate the value of this tangent using a calculator:
tan(angle) ≈ 0.8871
To find the angle itself, we can use the inverse tangent function (tan^(-1)):
angle ≈ tan^(-1)(0.8871)
Using a calculator, we find:
angle ≈ 43.6 degrees
Therefore, the angle of elevation of the sun is approximately 43.6 degrees.
To find the angle of elevation of the sun, we can use trigonometry. Let's draw a diagram to represent the situation:
/
/|
/ | <- shadow (6.2 ft)
/ |
/ |
/ |
-------|
| <- woman (5.5 ft)
In this diagram, let's assume that the woman's height represents the opposite side of the triangle, and the length of the shadow represents the adjacent side. The angle of elevation that we want to find is the angle between the ground and the line of sight from the woman's eyes to the top of her head.
Now, we can use the tangent function to find the angle of elevation. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the woman's height (5.5 ft), and the adjacent side is the length of the shadow (6.2 ft). So we have:
tan(angle) = opposite/adjacent
Substituting the given values:
tan(angle) = 5.5/6.2
To find the angle, we can take the inverse tangent (also known as arctan) of both sides:
angle = arctan(5.5/6.2)
Using a calculator, we find that:
angle ≈ 40.78 degrees
Therefore, the angle of elevation of the sun is approximately 40.78 degrees.