A tree growing on a hillside casts a 121-foot shadow straight down
the hill. Find the height of the tree (in feet) if the slope of the hill is 8 degrees
and the angle of elevation of the sun from the horizontal is 50degrees.
see whether you can figure how I developed the required formula for the height h of the tree:
h = 121(cos8°tan50° - sin8°) = 126 ft
Draw a diagram and figure out some of the distances.
To find the height of the tree, we can use trigonometric ratios based on the given angles.
Let's define the following:
- h: height of the tree
- s: length of the tree's shadow
- θ: angle of elevation of the sun from the horizontal
- α: slope of the hill
From the given information, we have:
s = 121 feet (length of the tree's shadow)
θ = 50 degrees (angle of elevation of the sun)
α = 8 degrees (slope of the hill)
We can visualize the situation as a right triangle, where the height of the tree (h) is the opposite side, the length of the shadow (s) is the adjacent side, and the angle of elevation (θ) is the acute angle.
Using the tangent ratio (opposite/adjacent) in this right triangle, we have:
tan(θ) = h / s
Rearranging the equation to solve for h gives:
h = s * tan(θ)
Plugging in the values we know:
h = 121 feet * tan(50 degrees)
Now, to find the height of the tree, we need to evaluate the tangent of 50 degrees and calculate the product with the length of the shadow.
Using a scientific calculator or trigonometric table, we find that tan(50 degrees) ≈ 1.1918.
Substituting this value back into the equation:
h ≈ 121 feet * 1.1918
Evaluating the expression:
h ≈ 144.2138 feet
Therefore, the height of the tree is approximately 144.2138 feet.