Problem 26 - Algebra 2

Question

GEOCRAPHY Mr. Cardona is a regional sales manager for a company in Florida. Tampa, Orlando, and Ocala outline his region. If a coordinate grid in which 1 unit =10 miles is placed over the map of Florida with Tampa at the origin, the coordinates of the three cities are \((0,0),(7,5),\) and \((2.5,10) .\) Estimate the area of his sales territory.

Step-by-Step Solution

Answer

The area of his sales territory is approximately 2875 square miles.

Step 1: Identify Coordinates

Identify the coordinates of the cities on the map. Tampa is at \((0,0)\), Orlando is at \((7,5)\), and Ocala is at \((2.5,10)\).

Step 2: Calculate the Area in Grid Units

We use the Shoelace Formula to calculate the area given the coordinates. For coordinates \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) the area is: \[ \text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_1 - (y_1x_2 + y_2x_3 + y_3x_1) \right| \].Substitute the coordinates, \((0,0),(7,5),(2.5,10)\):\[ \text{Area} = \frac{1}{2} \left| 0\times5 + 7\times10 + 2.5\times0 - (0\times7 + 5\times2.5 + 10\times0) \right| \]\[ = \frac{1}{2} \left| 70 - 12.5 \right| \]\[ = \frac{1}{2} \cdot 57.5 \]\[ = 28.75 \text{ grid units}^2 \].

Step 3: Convert Grid Units to Miles

Since 1 grid unit equals 10 miles, the area in square miles is \[ 28.75 \times 10^2 = 2875 \text{ square miles} \].

Key Concepts

Coordinate Geometry

Coordinate Geometry is a vital tool in studying geographical locations by using coordinates. These coordinates are points on a plane, expressed as pairs \(x, y\). Each number in the pair represents a location on the \(x\) (horizontal) and \(y\) (vertical) axes respectively.
This exercise places Tampa, Orlando, and Ocala on a coordinate grid over a map, using these principles:

• Tampa: Located at the origin (0,0) — where the x-axis and y-axis meet.
• Orlando: Located at (7,5) — 7 units along the x-axis and 5 units up the y-axis.
• Ocala: Positioned at (2.5,10) — 2.5 units on the x-axis and 10 units up the y-axis.

Understanding these coordinates helps us visualize and calculate distances, angles, and areas between these locations.

Shoelace Formula

The Shoelace Formula is a mathematical technique used to find the area of a polygon when given its vertex coordinates. It's named so because the calculation mimics the process of tying a shoelace, moving around the vertices in sequence.
To understand how it works, consider the formula for three points: \[\text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_1 - (y_1x_2 + y_2x_3 + y_3x_1) \right|\]Here, \(x_1, y_1\), \(x_2, y_2\), and \(x_3, y_3\) represent the coordinates of the triangle's vertices. The formula essentially calculates the area enclosed by these points by summing products of coordinates in a specific pattern. For Tampa, Orlando, and Ocala's triangle, this formula allows us to seamlessly compute their enclosed area.

Area Calculation

Calculating the area of a region, like between Tampa, Orlando, and Ocala, requires using the Shoelace Formula to establish the area in grid units. Here's a simplified breakdown:

• Substitute the given coordinates into the Shoelace Formula: Tampa \((0,0)\), Orlando \((7,5)\), and Ocala \((2.5,10)\).
• Perform the multiplications and make the necessary subtractions to consolidate terms. For instance, calculate \(70 - 12.5\) as part of the formula.
• Halve the result, as indicated by the formula — hence, \(28.75\) grid units squared represents the triangle's area.

These steps ensure a thorough calculation method, accommodating diverse polygons by adjusting the formula for more vertices if required.

Grid to Miles Conversion

Converting grid units to real-world measurements makes the area meaningful within a geographical context. Our grid uses 1 unit as equal to 10 miles, suitable for large distances like those between cities.
To convert the area from grid units squared to square miles, apply a multiplication factor considering the scale:

• If 1 unit equals 10 miles, 1 grid unit squared reflects \((10 \text{ miles}) \times (10 \text{ miles}) = 100 \text{ square miles}\).
• Multiply the computed area in grid units squared by 100. With an area of \(28.75 \text{ grid units}^2\), multiply to get \(2875 \text{ square miles}\).

This conversion clarifies the scale and extent of regions like a sales territory on a map of Florida, making calculations practical and useful for planning purposes.