Problem 25 - Prealgebra

Question

SELECT A TECHNIQUE In a golf tournament, Joan's ball landed 2 feet to the left and 3 feet short of the cup. Carolina's ball landed 1 foot to the right and 4 feet beyond the cup. Which of the following techniques would you use to determine who is closer to the cup? Justify your selection(s). Then use the technique(s) to solve the problem.

Step-by-Step Solution

Answer

Use the distance formula; Joan is closer to the cup.

Step 1: Identify the Positions Relative to the Cup

Joan's ball is 2 feet to the left and 3 feet short of the cup, which places her position at (-2, -3) in a coordinate plane where the cup is at the origin (0, 0). Carolina's ball is 1 foot to the right and 4 feet beyond the cup, positioning her at (1, 4).

Step 2: Choose the Technique for Comparison

To determine who is closer to the cup, we need to calculate the distance from each ball to the cup using the distance formula for points in a plane: \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). This formula will help in comparing both distances objectively.

Step 3: Calculate Joan's Distance to the Cup

Using the distance formula for Joan's position \((-2, -3)\) and the cup at \((0, 0)\):\[ d_1 = \sqrt{(-2 - 0)^2 + (-3 - 0)^2} = \sqrt{4 + 9} = \sqrt{13} \]

Step 4: Calculate Carolina's Distance to the Cup

Using the distance formula for Carolina's position \((1, 4)\) and the cup at \((0, 0)\):\[ d_2 = \sqrt{(1 - 0)^2 + (4 - 0)^2} = \sqrt{1 + 16} = \sqrt{17} \]

Step 5: Compare the Distances

After calculating the distances, Joan's distance is \(\sqrt{13}\) and Carolina's is \(\sqrt{17}\). Since \(\sqrt{13} < \sqrt{17}\), Joan's ball is closer to the cup than Carolina's.

Key Concepts

Coordinate Plane

A coordinate plane is a two-dimensional surface where we can represent points using a pair of numerical coordinates. This system helps to locate the position of points by specifying a horizontal position (along the x-axis) and a vertical position (along the y-axis). In this context, the coordinate plane provides a straightforward means to visually understand geometric positions and relationships.

When determining the location of points in problems like the golf tournament example, we assign one central point, often called the origin, that acts as a reference point. In this exercise, the location of the cup is set at the origin (0, 0). Each player's ball location is then described by how far left or right and how far above or below this origin point the ball landed.

• Joan's Ball Position: (-2, -3) - This signifies 2 feet leftwards and 3 feet downwards from the origin (cup).
• Carolina's Ball Position: (1, 4) - This signifies 1 foot to the right and 4 feet upwards from the origin (cup).

This type of exercise allows for a clear and organized approach in understanding and solving problems related to positions and distances.

Pythagorean Theorem

The Pythagorean Theorem is a powerful mathematical tool that connects the lengths of the sides in a right triangle. This fundamental concept in geometry states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Mathematically, it's expressed as:

• For a right triangle with sides of lengths 'a' and 'b', and hypotenuse 'c', the theorem is represented as:

\[ a^2 + b^2 = c^2 \]

In the context of determining distances on a coordinate plane, we can use the Pythagorean Theorem to calculate how far apart two points are. When you think about a single point on the plane, you can imagine a triangle formed by drawing a line horizontally from one point and vertically to the other. The diagonal between them represents the direct distance from point to point, which serves as the hypotenuse of our imagined triangle.

Thus, whenever you encounter a problem involving distance on a plane, the Pythagorean Theorem aids in deriving the Distance Formula, crucial for determining positions as we've seen with Joan's and Carolina's shots on the golf course.

Calculation Techniques in Math

In mathematics, having a good grasp of calculation techniques is essential to solve various problems effectively. In this exercise, one specific technique we use is the Distance Formula, which is derived from the Pythagorean Theorem.

The Distance Formula enables us to compute the length or distance between two points on a coordinate plane. Given two points,

• If the first point is \(x_1, y_1\), and the second point is \(x_2, y_2\), the formula is:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

This formula is paramount in our golf scenario. It helped us calculate how close each ball was to the cup:

• Joan's Distance: Using her coordinates (-2, -3), the distance computed was \(\sqrt{13}\).
• Carolina's Distance: With her coordinates (1, 4), the calculation resulted in \(\sqrt{17}\).

By comparing these results, we can conclude who is closer to the cup. Understanding and practicing these techniques allows students to tackle similar mathematical calculations with increased confidence and proficiency.