the leaning tower of pisa is 179 ft tall. it leans at an angle of 5.5 degrees. if the angle of elevation of the sun is 60 degrees and the sun is opposite the side on which the tower leans, how long is the shadow that the tower will cast?
Make a sketch.
I have a triangle ABC, where BC is the ground, AC is the tower.
Angle B = 60°, angle C = 84.5°, making angle A = 35.5°
by Sine Law:
BC/sin 35.5° = 179/sin 60°
I get BC = 120.026
So the shadow of the tower is 120 ft long
the angle of elevation of the sun is 31 degrees. find the lenght of the shadow, to the nearest foot, of a man that is 6feet tall.
To find the length of the shadow that the tower will cast, you can use trigonometry.
First, let's define the given information:
- Height of the tower (h) = 179 ft
- Angle of inclination (θ) = 5.5 degrees
- Angle of elevation of the sun (α) = 60 degrees
Now, we need to calculate the length of the shadow (S).
Step 1: Calculate the height of the tower that is perpendicular to the ground.
Since the tower is leaning, we can use the trigonometric relationship:
Sin(θ) = Opposite / Hypotenuse
In this case, the opposite side is the perpendicular height of the tower (h' = ??), and the hypotenuse is the overall height of the tower (h = 179 ft).
Rearranging the formula, we get:
h' = h * Sin(θ)
h' = 179 ft * Sin(5.5°)
Calculate h':
h' = 179 ft * 0.096 <- Calculated using a calculator
h' ≈ 17.184 ft
So, the height of the tower perpendicular to the ground is approximately 17.184 ft.
Step 2: Calculate the length of the shadow (S).
Now, we have a right triangle formed between the tower, the shadow, and the angle of elevation of the sun.
We know the height of the tower perpendicular to the ground (h') and the angle of elevation of the sun (α = 60°).
Tan(α) = Opposite / Adjacent
In this case, the opposite side is the height of the tower (h'), and the adjacent side is the length of the shadow (S).
Rearranging the formula to solve for S:
S = h' / Tan(α)
S = 17.184 ft / Tan(60°)
Calculate S:
S = 17.184 ft / 1.732 <- Calculated using a calculator
S ≈ 9.92 ft
Therefore, the shadow that the leaning tower of Pisa will cast is approximately 9.92 ft long.