The leaning tower of Pisa is 55.9 m tall and leans 5.5 degrees from the vertical. If its shadow is 90 m long, what is the distance from the top of the tower to the top of the edge of its shadow, assuming the ground around the tower is level?
oh yeah that's true where does the 84.5 degrees come from?
the tower makes an angle of 84.5º with the ground.
let x m be the length of the shadow, then by the Cosine Law
x^2 = 90^2 + 55.9^2 - 2(90)(55.9)cos 84.5º
etc.
where did you get the 84.5º from?
thanks so much!
A vertical makes a right-angle with the horizontal
The tower is off 5.5º from the vertical.
5.5º + ? = 90 ??
84.5 degrees:
90+5.5=95.5
180-95.5=84.5
To find the distance from the top of the tower to the top of the edge of its shadow, we can use trigonometry. Let's assume that the distance we are looking for is 'x'.
We have the following information:
- Height of the tower (h) = 55.9 m
- Angle of inclination (θ) = 5.5 degrees
- Length of the shadow (s) = 90 m
We are trying to find the distance from the top of the tower to the top of the edge of its shadow, which is given by h + x.
To approach this problem, we can use the tangent function (tanθ), which relates the angle of inclination to the height and the distance of the shadow.
The tangent of the angle of inclination (θ) is defined as the ratio of the opposite side (height of the tower) to the adjacent side (distance of the shadow). In mathematical terms, we can write this as:
tan(θ) = h / s
Rearranging this equation to solve for h, we get:
h = s * tan(θ)
Now, substituting the given values into the equation, we have:
h = 90 * tan(5.5°)
Using a scientific calculator, we can evaluate tan(5.5°), which is approximately 0.096.
h = 90 * 0.096
≈ 8.64 m
So, the height from the top of the tower to the top of the edge of its shadow is approximately 8.64 meters.