Problem 46 - Precalculus : Building Concepts and Connections

Question

Games A billiard ball traverses a distance of 26 inches on a straight-line path, and then it collides with another ball, changes direction, and traverses a distance of 18 inches on a different straight-line path before coming to a stop. If an angle of \(37^{\circ}\) is formed from the lines that connect the initial location of the ball to the final location of the ball and to the point of the collision, what are the two possible values of the distance \(d\) between the initial and final locations of the ball? Sketch a figure first.

Step-by-Step Solution

Answer

By solving the derived equation, the two possible lengths for the distance \(d\), or the two possible distances between the initial and final locations of the ball, can be obtained.

Step 1: Sketch the Triangle

Draw a triangle with \(26\) inches and \(18\) inches as two sides and a \(37^{\circ}\) included angle (angle between the initial and altered path). Mark the initial point, collision point and final point. The line connecting the initial and final point represents the distance \(d\) we want to find.

Step 2: Apply the Law of Cosines

The law of cosines is applied to relate the sides and included angle of a triangle and can be written as \(c^2 = a^2 + b^2 - 2ab \cdot cos(C)\), where \(c\) is the side opposing the angle \(C\) and \(a\) and \(b\) are the other two sides. Substitute \(a = 26\), \(b = 18\), and \(C = 37^{\circ}\) to calculate \(c\) or \(d\). Using the cosine value of \(37\), we get \(d^2 = 26^2 + 18^2 - 2 \cdot 26 \cdot 18 \cdot cos(37^{\circ})\).

Step 3: Calculate the Value of \(d\)

The calculation in the previous step leaves us with a quadratic equation in \(d\) which could be solved to give two possible values \(d_1\) and \(d_2\). Evaluating the expression using appropriate number values would provide the solutions for \(d\). Make sure to evaluate the square root correctly to get both the positive and negative roots.

Step 4: Interpretation of Results

Two values for \(d\) would represent two possible scenarios. The lengths could be the distances from where the billiard ball started to where it ended up before and after the bounce respectively depending on how the triangle is interpreted.

Key Concepts

Trigonometry and its Role in Solving the Problem

Trigonometry is the mathematical study of relationships involving the lengths and angles of triangles. It plays a significant role in solving problems related to angles and distances. In our billiard ball problem, trigonometry helps us determine the unknown distance between the initial and final positions of the ball. This is achieved through the use of the Law of Cosines, an essential tool in trigonometry that allows us to calculate the side lengths of a triangle when two sides and the included angle are known.

This law is typically expressed as:

• \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \)

In this formula, \( c \) represents the side opposite angle \( C\), and \( a \) and \( b \) correspond to the other two sides. Utilizing this equation, we substituted the given values of the triangle's sides and the angle, leading us to the necessary calculations for achieving the solution.

Problem-Solving Approach

When addressing a geometry problem, especially one involving real-life applications such as our billiard ball scenario, a structured problem-solving approach becomes essential. In this exercise, we've systematically

• sketched the situation, creating a visual representation of the triangle involved,
• applied the trigonometric laws, in particular, the Law of Cosines,
• translated the physical problem into a mathematical equation,
• calculated the unknown side length, and
• interpreted the results to deduce the implications on the billiard ball's trajectory.

Breaking down a complex problem into smaller, more manageable steps helps in clarifying what is needed to find the solution. This helps avoid confusion and ensures all aspects of the problem are considered.

Geometry is Fundamental in Understanding the Exercise

Geometry involves the study of shapes, sizes, and the properties of space. In this problem, geometry comes into play as we visually map out the trajectory of a billiard ball. The task involves understanding how to

• represent the ball’s path using lines,
• form triangles that relate to real-world distances, and
• use the angle between these lines to gather further information about the paths taken.

By drawing a triangle with specific sides such as 26 inches and 18 inches and an included angle of \( 37^{\circ} \), students can physically see the parameters they are working with. This helps in connecting the abstract idea of mathematical equations with the visual layout presented in the exercise.

Understanding Billiard Trajectory

In a game of billiards, a player's control over the ball's path is crucial. The ball's trajectory can be understood and predicted using principles from physics and mathematics. In our problem, once the billiard ball collides and changes direction, it forms a new path at an angle. By analyzing this, students see how geometry and trigonometry can explain and solve real-world physics phenomena.

When we calculate different potential outcomes for the distance \(d\), we are effectively predicting how different angles and encounters with objects (like billiard balls) change the ball's motion. This understanding helps in strategizing during gameplay, knowing which angles yield beneficial outcomes. Mastering these concepts not only assists in solving academic exercises but also enhances practical skills on the billiard table.