Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Stonehenge blocks Stonehenge in Salisbury Plains, England, was constructed using solid stone blocks weighing over 99,000 pounds each. Lifting a single stone required 550 people, who pulled the stone up a ramp inclined at an angle of \(9^{\circ}\). Approximate the distance that a stone was moved in order to raise it to a height of 30 feet.

Step-by-Step Solution

Answer

The stone was moved approximately 191.85 feet.

Step 1: Understanding the Problem

We need to determine the distance a stone was moved up a ramp to reach a height of 30 feet, given the ramp's incline angle of 9 degrees.

Step 2: Identify the Triangle

The situation can be visualized as a right triangle, where the height (30 feet) is the opposite side, the hypotenuse is the distance the stone was moved, and the angle is 9 degrees.

Step 3: Apply the Sine Function

The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. Therefore, we have: \( \sin(9^{\circ}) = \frac{30}{d} \) where \( d \) is the distance the stone was moved.

Step 4: Solve for the Hypotenuse

Rearrange the equation to solve for \( d \):\[ d = \frac{30}{\sin(9^{\circ})} \]

Step 5: Calculate the Distance

Calculate \( \sin(9^{\circ}) \) using a calculator, which gives approximately 0.1564. Now substitute this into the equation to find \( d \):\[ d \approx \frac{30}{0.1564} \approx 191.85 \] feet.

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This means that the triangle has one right angle, making the two sides that meet at this angle perpendicular. In the context of Stonehenge and its construction, understanding right triangles is crucial. The incline ramp used to move the heavy stones can be simplified into a right triangle model.
In a right triangle, you have three sides:

• The hypotenuse: This is the longest side, opposite the right angle.
• The opposite side: This is the side opposite to the angle of interest.
• The adjacent side: This lies next to the angle of interest and forms the right angle with the opposite side.

In our problem, the height the stones were raised to represents the opposite side, while the hypotenuse is the distance the stone traveled along the inclined plane. Recognizing these elements helps us to manipulate mathematical functions to solve real-world problems.

Sine Function

The sine function is a fundamental concept in trigonometry. It relates an angle in a right triangle to the ratio of two of its sides: the opposite and the hypotenuse. For an angle \( \theta \), the sine function is expressed as:

• \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)

In the case of the Stonehenge block problem, the angle is \(9^{\circ}\), the opposite side is the height of 30 feet, and the hypotenuse is the distance we need to find. By rearranging the sine equation, you can solve for the hypotenuse. This involves dividing the length of the opposite side by the sine of the angle:

• \( d = \frac{30}{\sin(9^{\circ})} \)

This allows you to calculate the total distance the stone was moved along the incline.

Inclined Plane

An inclined plane is a flat surface tilted at an angle, other than horizontal. It allows for the movement of objects along with it, often making it easier to move something upwards by decreasing the amount of force needed as compared to lifting it vertically. It applies basic mechanical principles and is a common tool documented throughout early human history.
To visualize, imagine a ramp which essentially is an inclined plane. In our context, the ramp helped move the heavy blocks to their needed height. The angle of inclination (which is \(9^{\circ}\) in this scenario) determines how steep the ascent. The properties of the inclined plane transform this physical setup into the right triangle model discussed earlier. This way, mathematical principles of triangles simplify calculations.

Angle Measurement

Understanding how angles are measured is essential in applying trigonometry. Angles are typically measured in degrees or radians, but in many problems, such as our Stonehenge example, degrees are used. A full circle is \(360^{\circ}\), and each degree represents \(1/360\)th of a full turn.
The given problem involves an angle of \(9^{\circ}\), which might seem small but has a massive impact. Small changes in angle can greatly alter the calculations due to their placement in a trigonometric function.
It's crucial to have an accurate measurement of angles for precise calculations, especially in engineering and physics, where incorrect calculations can lead to errors. Tools like protractors, inclinometers, or digital angle gauges are often used to measure these angles. The sine function directly translates these angle measurements into useful dimensions needed for solving real-world problems.