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 (see figure). The top of the tower leans about 5.45 meters off center.
Use the Law of Sines to write an equation for the length d of the shadow cast by the tower in terms of θ.
1/pi
To write an equation for the length of the shadow cast by the tower in terms of θ using the Law of Sines, we need to set up a triangle. Let's consider the following figure:
|
|
| θ
|
o----------x----------o
/ | /
d / | / h
/_____________| /
b
In the triangle above, the line segment "ox" represents the tower, and "oh" represents the height of the tower, which is 58.36 meters. The length of the shadow cast by the tower is denoted as "d." The angle between the length of the shadow and the height of the tower is represented as "θ," and the base of the triangle is denoted as "b." We want to find the equation for "d" in terms of "θ."
According to the Law of Sines, we have the following relationship:
sin(θ) / b = sin(90°) / h
Since sin(90°) equals 1, the equation can be simplified as follows:
sin(θ) / b = 1 / h
Now, we can rearrange the equation to solve for "d" in terms of "θ." Since sin(θ) = h / x using basic trigonometric ratios, we can substitute this value into the equation:
(h / x) / b = 1 / h
Simplifying further,
h^2 = xb
Finally, to solve for "d," we substitute the given values into the equation:
d = x - b
Therefore, the equation for the length of the shadow cast by the tower in terms of θ is:
d = x - b = √(h^2 + b^2) - b
where h = 58.36 meters and b = 5.45 meters.
To use the Law of Sines to write an equation for the length of the shadow cast by the tower in terms of θ, we need to consider the given information. The tower leans 5.45 meters off center, which means we can assume that the shadow cast by the tower is perfectly perpendicular to the ground.
Let's denote the length of the shadow as d and the angle of elevation from the top of the tower to the tip of the shadow as θ. Now, we can set up the equation using the Law of Sines.
According to the Law of Sines, the ratio of a side length to the sine of its respective opposite angle in a triangle is constant. In this case, the side length is the length of the shadow d, the opposite angle is θ, and the opposite side is the height of the tower h.
Therefore, we can write the equation as:
d/sin(θ) = h/sin(90°)
Since the sine of 90° is equal to 1, the equation simplifies to:
d/sin(θ) = h
Substituting the given values, we have:
d/sin(θ) = 58.36
To isolate d, we can multiply both sides of the equation by sin(θ):
d = 58.36 * sin(θ)
Thus, the equation for the length d of the shadow cast by the tower in terms of θ is:
d = 58.36 * sin(θ)