The leaning Tower of Pisa in Italy leans because it was built on unstable soil – a mixture of clay, sand, and water.
The tower is approximately 58.36 meters tall from its foundation. The top of the tower leans about 5.45 meters
off center. Find the angle of lean of the tower
as usual, draw a diagram and review your basic trig functions.
sinθ = 5.45/58.36
To find the angle of lean of the Tower of Pisa, we can use trigonometry.
First, we need to determine the length of the horizontal distance from the top of the tower to the center of its base.
Let's call the height of the tower "h" and the horizontal distance "d". Given that the top of the tower leans about 5.45 meters off center, we can say that d = 5.45 meters.
Using the Pythagorean theorem, we can find the length of the slanted side of the tower, which we'll call "s". The equation for the Pythagorean theorem is:
s^2 = h^2 + d^2
Plugging in the known values, we have:
s^2 = (58.36 meters)^2 + (5.45 meters)^2
Simplifying,
s^2 = 3406.4096 meters^2 + 29.7025 meters^2
s^2 = 3436.1121 meters^2
Taking the square root of both sides,
s = √(3436.1121 meters^2)
s ≈ 58.6239 meters
Now we can calculate the angle of lean, θ, using the tangent of the angle. The tangent is defined as the ratio of the opposite side (h) to the adjacent side (d) in a right triangle. Therefore,
tan(θ) = h / d
Plugging in the values we have:
tan(θ) = 58.36 meters / 5.45 meters
tan(θ) ≈ 10.7184
To find the angle, we need to take the inverse tangent (also known as arctan) of tan(θ):
θ = arctan(10.7184)
θ ≈ 83.4768 degrees
Therefore, the angle of lean of the Tower of Pisa is approximately 83.4768 degrees.
To find the angle of lean of the Tower of Pisa, we can use trigonometry and the given measurements.
First, let's define some variables for better understanding:
H = Height of the tower (58.36 meters)
D = Horizontal distance from the center of the tower to the point directly below the top (5.45 meters)
Now, to find the angle of lean, we can use the tangent function (tan):
tan(angle) = D/H
Substituting the values:
tan(angle) = 5.45/58.36
To find the angle itself, we need to take the inverse tangent (arctan) of both sides:
angle = arctan(5.45/58.36)
Now, we can calculate the angle by either using a scientific calculator or an online calculator that provides trigonometric functions.
Once you input the expression "arctan(5.45/58.36)" into the calculator, it will give you the angle of lean of the Tower of Pisa.