Problem 5 - Algebra 2

Question

CELL PHONES A person using a cell phone can be located in respect to three cellular towers. In a coordinate system where a unit represents one mile, the caller is determined to be 50 miles from a tower at the origin, 40 miles from a tower at \((0,30),\) and 13 miles from a tower at \((35,18) .\) Where is the caller?

Step-by-Step Solution

Answer

The caller is at the coordinates (40, 30).

Step 1: Write the equation for each tower

We will use the distance formula to write an equation for the caller's distance from each tower. The distance formula for a point \((x, y)\) from a tower at \((x_1, y_1)\) is given by \(\sqrt{(x-x_1)^2 + (y-y_1)^2}\).

Step 2: Set equations for Tower 1 (at origin)

The caller is 50 miles from the origin tower. Using the distance formula, the equation is:\[x^2 + y^2 = 50^2\]This simplifies to \(x^2 + y^2 = 2500\).

Step 3: Set equations for Tower 2 (at (0, 30))

The caller is 40 miles away from the tower at \((0, 30)\). Using the distance formula, the equation is:\[(x-0)^2 + (y-30)^2 = 40^2\]This simplifies to \(x^2 + (y-30)^2 = 1600\).

Step 4: Set equation for Tower 3 (at (35, 18))

The caller is 13 miles away from the tower at \((35, 18)\). Using the distance formula, the equation is:\[(x-35)^2 + (y-18)^2 = 13^2\]This simplifies to \((x-35)^2 + (y-18)^2 = 169\).

Step 5: Solve the equations simultaneously

We have three equations:1. \(x^2 + y^2 = 2500\)2. \(x^2 + (y-30)^2 = 1600\)3. \((x-35)^2 + (y-18)^2 = 169\)First, simplify the second equation:\(x^2 + y^2 - 60y + 900 = 1600\), then substituting \(x^2 + y^2 = 2500\) gives\(2500 - 60y + 900 = 1600 \rightarrow 3400 - 60y = 1600 \rightarrow 60y = 1800 \rightarrow y = 30\).Substitute \(y = 30\) back into the first equation to solve for \(x\):\(x^2 + 30^2 = 2500 \rightarrow x^2 + 900 = 2500 \rightarrow x^2 = 1600 \rightarrow x = 40\) or \(x = -40\).To check with the third equation:\((40-35)^2 + (30-18)^2 = 25 + 144 = 169\), which is true.Thus, the caller is at \((40, 30)\).

Step 6: Solution verification

Verify that the points satisfy all three equations derived from the distances: For tower 1: \(40^2 + 30^2 = 1600 + 900 = 2500\),For tower 2: \(40^2 + (30-30)^2 = 1600\),For tower 3: \((40-35)^2 + (30-18)^2 = 25 + 144 = 169\).All equations are satisfied, confirming the solution.

Key Concepts

Equation Solving

Solving equations in coordinate geometry often involves using the Distance Formula. This formula allows us to determine the distance between two points in a coordinate plane. To find a caller's location relative to cellular towers, we start by defining an equation for each tower based on the caller's distance.
The Distance Formula is described as:\[ \sqrt{(x-x_1)^2 + (y-y_1)^2} = d \] where \((x_1, y_1)\) represents the coordinates of a tower, and \(d\) is the distance from the caller to the tower. The goal is to express this relationship in a way that it can be solved mathematically. By squaring both sides of the equation, you can eliminate the square root, making it easier to handle algebraically:

• For Tower 1, solve: \(x^2 + y^2 = 2500\)
• For Tower 2, solve: \(x^2 + (y-30)^2 = 1600\)
• For Tower 3, solve: \((x-35)^2 + (y-18)^2 = 169\)

Mastering how to derive and manipulate these equations is a critical step in finding the caller's exact location.

Simultaneous Equations

Simultaneous equations are sets of equations that are solved together because they share variables. In our distance calculation problem, we have three such equations:

• \(x^2 + y^2 = 2500\)
• \(x^2 + (y-30)^2 = 1600\)
• \((x-35)^2 + (y-18)^2 = 169\)

The objective is to find common solutions for \(x\) and \(y\) that satisfy all equations. First, simplify one of the equations or use algebraic manipulation to solve for one variable in terms of the other.
For instance, in this case:

• Rearrange the second equation to: \(2500 - 60y + 900 = 1600\) which simplifies to \(60y = 1800\) and \(y = 30\).

With \(y\) known, substitute back into another equation, like \(x^2 + 30^2 = 2500\), allowing you to solve for \(x\). Repeat this process until all variables are determined. This systematic approach ensures the location is accurate.

Geometric Location

The geometric location of a point in coordinate geometry refers to identifying the specific coordinates that meet given constraints, such as being at certain distances from known points. In this exercise, the aim is to pinpoint where the cell phone user is located based on their distance from three cellular towers.
Here's the process:

• You're given the distances from the towers and their coordinates.
• Using the Distance Formula, set equations that represent the caller's distance from each tower, creating a system of equations.
• Solve these simultaneous equations to find the coordinates \((x, y)\) where all conditions are satisfied.

This exercise vividly illustrates the concept of triangulation in navigation. Once solved, the caller's position is confirmed to be at \((40, 30)\). Understanding how different equations intersect and define a unique location on the coordinate plane is fundamental in coordinate geometry, proving invaluable in applications such as GPS technology.