Problem 91 - Algebra and Trigonometry

Question

Stonehenge, the famous "stone circle" in England, was built between 2750 B.C. and 1300 B.C. using solid stone blocks weighing over \(99,000\) pounds each. It required 550 people to pull a single stone up a ramp inclined at a \(9^{\circ}\) angle. Describe how right triangle trigonometry can be used to determine the distance the 550 workers had to drag a stone in order to raise it to a height of 30 feet.

Step-by-Step Solution

Answer

The workers had to drag the stone over a distance of approximately 191.18 feet to raise it to a height of 30 feet.

Step 1: Identify Given Variables

In this case, the given variables are: the angle of the ramp, \(\theta = 9^{\circ}\), and the height, \(h = 30\) feet.

Step 2: Apply Right Triangle Trigonometry

Since the height is the opposite side and the distance the workers had to drag the stone, the hypotenuse, you will use the sine function. This function, for a given angle \(\theta\), is defined as the opposite divided by the hypotenuse, or \(\sin(\theta) = \frac{h}{d}\), with \(d\) is the unknown distance.

Step 3: Solve for the Unknown Variable

The equation from Step 2 can be rearranged to solve for the unknown distance: \(d = \frac{h}{\sin(\theta)}\)

Step 4: Substitute with the Given Variables and Calculate

Substitute the given height and angle into the equation: \(d = \frac{30}{\sin(9^{\circ})}\) After calculation, you will get the distance that the workers had to drag the stone.

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are fundamental tools in mathematics, especially when dealing with triangles. These functions relate the angles of a triangle to the lengths of its sides. The main trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). Each function provides a way to calculate a specific ratio of the sides of a right triangle.

For example, the sine function is defined as the ratio of the length of the opposite side to the hypotenuse. Similarly, cosine is the ratio of the adjacent side to the hypotenuse, and tangent combines both concepts as the ratio of the opposite side to the adjacent side. These relationships are incredibly useful in solving problems involving angles and distances.

• Simplifies calculating unknown sides
• Essential in navigation, physics, and engineering

Exploring the Sine Function

The sine function is a crucial part of right triangle trigonometry. It helps you determine unknown measurements when certain details are given, like an angle and a side length.

Consider a right triangle where you know the height (opposite side) and angle inclination; you can find the hypotenuse — the side opposite the right angle — using the sine function.

In this Stonehenge ramp problem, the sine function is perfect because it specifically relates the angle of ramp inclination and height (distance workers raised the stone) to the unknown length of the ramp (hypotenuse). This is expressed in the equation:\[\sin(\theta) = \frac{h}{d}\]where:

• \(\theta\) is the given angle
• \(h\) is the given height (30 feet)
• \(d\) is the distance

Solving for \(d\) helps you figure out how far the workers had to drag the stone.

Approaches to Mathematical Problem-Solving

Effective problem-solving in mathematics relies on identifying given data, selecting the appropriate method, and calculating accurately. Breaking down a problem into manageable steps can greatly enhance comprehension and solution accuracy.

Begin by identifying key variables and information, such as specific angles or sides in right triangle problems. Applying relevant mathematical concepts like trigonometric functions allows you to craft equations that reflect these relationships.

For the Stonehenge example, the correct application of the sine function was necessary for calculating the ramp's hypotenuse. Then, rearranging the equation and substituting known values let us derive the solution effectively. When you follow these steps, even complex problems become easier and more accessible. Remember to re-check calculations to ensure they are correct and adjust if necessary. This systematic approach is vital in academic and practical applications.