Problem 35 - Contemporary Precalculus

Question

A visitor to the Leaning Tower of Pisa observed that the tower's shadow was 40 meters long and that the angle of elevation from the tip of the shadow to the top of the tower was \(57^{\circ} .\) The tower is now 54 meters tall (measured from the ground to the top along the center line of the tower). Approximate the angle \(\alpha\) that the center line of the tower makes with the vertical. (IMAGES CANNOT COPY)

Step-by-Step Solution

Answer

Answer: To find the angle α, follow the steps below: 1. Calculate the length of the ground from the shadow's tip to the top of the tower using the tangent function: opposite = tan(57°) × 40. 2. Calculate the length of the centerline (hypotenuse) using the Pythagorean theorem: (opposite)² + (40)² = (centerline)². 3. Calculate the angle α using the sine function: α = arcsin(opposite / centerline). After performing the calculations, you will find the angle α that the centerline of the tower makes with the vertical axis.

Step 1: Find the length of the ground to the top of the tower's shadow (opposite side)

We have the angle of elevation \(57^{\circ}\) and the shadow length (40 meters), so we can use the tangent function to find the length of the ground from the shadow's tip to the top of the tower: \(\tan(57^{\circ}) = \frac{opposite}{adjacent}\) We have the adjacent side, which is the shadow length of 40 meters. So, we can find the opposite side (the height from the ground to the top of the tower's shadow) by multiplying both sides by the adjacent side: \(opposite = \tan(57^{\circ}) \times 40\) Calculate the length of the opposite side.

Step 2: Find the length of the centerline of the tower (hypotenuse)

Now that we have the opposite side, we can use the Pythagorean theorem to find the length of the centerline (the hypotenuse): \(a^2 + b^2 = c^2\) The opposite side, found in Step 1, is a, and the adjacent side is the shadow's length (40 meters). We need to find c (the length of the centerline): \((opposite)^2 + (40)^2 = (centerline)^2\) Calculate the length of the centerline (hypotenuse).

Step 3: Find the angle \(\alpha\) that the centerline makes with the vertical

Now that we have the centerline's length (hypotenuse) and the tower's height (opposite side), we can use the sine function to find angle \(\alpha\): \(\sin(\alpha) = \frac{opposite}{hypotenuse}\) We know the length of the opposite side (found in Step 1) and the hypotenuse (found in Step 2): \(\sin(\alpha) = \frac{opposite}{centerline}\) Calculate the angle \(\alpha\) with its arc sine (inverse sine) function: \(\alpha = \arcsin{\left(\frac{opposite}{centerline}\right)}\) Finally, calculate the angle \(\alpha\) to find the angle that the centerline of the tower makes with the vertical.

Key Concepts

Trigonometric Functions

Understanding trigonometric functions is essential when it comes to solving problems involving right-angled triangles. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan), each representing a relationship between two sides of a right triangle in relation to one of its acute angles.

For instance, the sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse (the side opposite the right angle). Similarly, the cosine relates the adjacent side to the hypotenuse, and the tangent is the ratio of the opposite side to the adjacent side. In mathematical terms:

• \(\text{sin(angle)} = \frac{\text{opposite}}{\text{hypotenuse}}\)
• \(\text{cos(angle)} = \frac{\text{adjacent}}{\text{hypotenuse}}\)
• \(\text{tan(angle)} = \frac{\text{opposite}}{\text{adjacent}}\)

In the Leaning Tower of Pisa problem, we utilize the tangent function to relate the length of the tower's shadow (the adjacent side) to the height of the tower above ground (the opposite side) given the angle of elevation. By understanding these relationships, students can more easily navigate problems involving angles and lengths in right triangles.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, especially useful for right-angled triangles. It states that in any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The theorem is generally expressed as:

• \(a^2 + b^2 = c^2\)

Where \(a\) and \(b\) are the lengths of the legs (the sides that meet at the right angle), and \(c\) is the length of the hypotenuse.

In the context of the Leaning Tower of Pisa exercise, once you have determined the length of the ground to the top of the tower's shadow using the tangent function, you can apply the Pythagorean theorem to find the length of the centerline of the tower (the hypotenuse of the triangle formed by the tower, its shadow, and the line from the top of the shadow to the top of the tower). This understanding makes it possible to then solve for the angle the centerline makes with the vertical.

Inverse Trigonometric Functions

When solving for angles in right triangles, the need often arises to perform the inverse operation of the trigonometric functions, known as inverse trigonometric functions. These functions are

• arc sine (\(\arcsin\))
• arc cosine (\(\arccos\))
• arc tangent (\(\arctan\))

Each of these functions uses the ratio of sides to calculate the measure of an angle. For instance, the inverse sine or arc sine function (\(\arcsin\)) allows you to find the angle whose sine is a given number.
For example, if \(\alpha\) is an angle in a right triangle where \(\sin(\alpha) = \frac{opposite}{hypotenuse}\), then \(\alpha = \arcsin{\left(\frac{opposite}{hypotenuse}\right)}\).

In our exercise with the Leaning Tower of Pisa, after calculating the lengths of the necessary sides of the triangle, the angle \(\alpha\) that the centerline of the tower makes with the vertical is found using the inverse sine function. This step is what allows us to move from the ratio of the sides back to the angle measurement, completing the problem and providing the needed angle.