Problem 25 - Precalculus: Functions and Graphs

Question

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 191.76 feet along the ramp.

Step 1: Analyzing the Problem

We are given the weight of the stone, the angle of the ramp, and the height it was raised. Our task is to find the distance the stone was moved along the ramp, equivalent to the hypotenuse in the right triangle formed by the height and the inclined distance.

Step 2: Understanding Right Triangle Trigonometry

In the context of this problem, the ramp's height represents the opposite side, the ramp's inclined distance is the hypotenuse, and the angle of elevation is given. We'll use the sine function because it connects these quantities: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).

Step 3: Setting Up the Equation

We know from trigonometry that: \[ \sin(9^{\circ}) = \frac{30}{d} \] where \( d \) is the distance along the ramp that we want to calculate.

Step 4: Solving for the Hypotenuse

Rearrange the equation to solve for \( d \): \[ d = \frac{30}{\sin(9^{\circ})} \]. Using a calculator, find \( \sin(9^{\circ}) \), which is approximately 0.1564.

Step 5: Calculating the Distance

Now, divide the height by the sine of the angle to find the distance along the ramp: \[ d \approx \frac{30}{0.1564} \approx 191.76 \text{ feet} \].

Key Concepts

Right Triangle

A right triangle is a geometric shape with one of its angles measuring precisely 90 degrees. This is called the right angle, and it creates a unique relationship between the triangle’s sides that is invaluable in trigonometry. In our Stonehenge problem, the ramp, the ground, and the height the stone block was lifted form a right triangle. Here, the angle of elevation (9 degrees) and the height (30 feet) are critical components. The longest side opposite the right angle is called the hypotenuse. Understanding the properties of right triangles allows us to apply trigonometric functions to solve for unknown sides, thus revealing distances in real-world applications, such as in the construction of ancient structures.

Sine Function

The sine function is a key concept in trigonometry that helps us understand relationships between angles and sides in a right triangle. It is particularly useful when you know an angle and one side of the triangle and need to find another side. The sine function is defined as the ratio of the length of the side opposite to the angle to the length of the hypotenuse. Mathematically, it is expressed as:\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]In the context of the Stonehenge problem, the sine function is used to relate the 9-degree angle of elevation with the 30-foot vertical rise of the stone. By plugging these values into the sine formula, you can solve for the ramp’s hypotenuse, which gives the distance along the ramp.

Angles of Elevation

The angle of elevation is an angle formed by the horizontal line of sight and the line of sight up to an object. When you look up at something, the angle your line of sight makes with the horizontal plane is the angle of elevation. In trigonometry, this concept is pivotal in problems involving heights and distances.

• In a right triangle, the angle of elevation is measured from the horizontal up to the hypotenuse.
• It's always measured from the ground, moving upwards.

In the Stonehenge problem, the 9-degree angle represents the angle of elevation of the ramp. This angle helps us apply trigonometric functions, like sine, effectively to calculate other dimensions of the triangle, such as the hypotenuse.

Hypotenuse Calculation

Calculating the hypotenuse in a right triangle is a common application of trigonometric functions. The hypotenuse is always opposite the right angle and is the longest side of a right triangle. To find the hypotenuse, you typically need to know the values of the other sides or an angle and one side.

The Calculation Process

By using the sine function, the hypotenuse can be extracted from the equation:\[ d = \frac{\text{opposite}}{\sin(\theta)} \]In our problem involving a 9-degree ramp, the hypotenuse is what we're solving for by rearranging the equation to find:\[ d = \frac{30}{\sin(9^{\circ})} \]Calculating this using a calculator shows that the hypotenuse (the distance along the ramp) is about 191.76 feet, demonstrating the practical utility of trigonometry in engineering and archaeology contexts.