A rock dropped from the top of the Learning Tower of Pisa falls to a point 14 feet from its base. If the tower is 182 feet tall, at what angle does it lean at the ground?

Question

A rock dropped from the top of the Learning Tower of Pisa falls to a point 14 feet from its base. If the tower is 182 feet tall, at what angle does it lean at the ground?

Answer 1

what is arccos (14/182)

arc cos means the angle whose cosine is ...

Some nowadays use the nomenclature cos^-1 , but that is just a little to symbolee for me.

Answer 2

To find the angle at which the Learning Tower of Pisa leans at the ground, we can use trigonometry. We will use the tangent function, which is defined as the ratio of the opposite side to the adjacent side of a right triangle.

In this case, the opposite side is the height of the tower (182 feet) and the adjacent side is the distance the rock falls from the base (14 feet). Let's call the angle we want to find θ.

Using the tangent function, we can set up the following equation:

tan(θ) = opposite/adjacent
tan(θ) = 182/14

Now we can find the value of θ by taking the arctangent (inverse tangent) of both sides of the equation. This will give us the angle in radians:

θ = arctan(182/14)

Using a calculator or a trigonometric table, we can find that the arctangent of 182/14 is approximately 85.14 degrees.

Therefore, the angle at which the Learning Tower of Pisa leans at the ground is approximately 85.14 degrees.

Answer 3

To find the angle at which the tower leans at the ground, we can use trigonometry. We know the height of the tower (182 feet) and the horizontal distance the rock falls from the base (14 feet).

Let's call the angle we're looking for θ.

First, we need to draw a right triangle to represent the situation. The height of the triangle corresponds to the tower (182 feet), the horizontal base is the distance traveled by the rock (14 feet), and the hypotenuse is the distance from the top of the tower to the rock's landing point.

Now we can use the tangent function to find the angle:

tan(θ) = Opposite / Adjacent

In our case, the opposite side is the height of the tower (182 feet) and the adjacent side is the base distance (14 feet).

tan(θ) = 182 / 14

Now, we can find the value of θ by taking the inverse tangent (arctan) of both sides of the equation.

θ = arctan(182 / 14)

Calculating this on a calculator or using a trigonometric table, we find that θ is approximately 85.74 degrees.

Therefore, the tower leans at an angle of approximately 85.74 degrees at the ground.