Problem 63 - Algebra and Trigonometry with Analytic Geometry

Question

Locating a ship A ship is traveling a course that is 100 miles from, and parallel to, a straight shoreline. The ship sends out a distress signal that is received by two Coast Guard stations A and B, located 200 miles apart, as shown in the figure. By measuring the difference in signal reception times, it is determined that the ship is 160 miles closer to \(\mathrm{B}\) than to \(\mathrm{A}\). Where is the ship?

Step-by-Step Solution

Answer

The ship is located approximately 180 miles from station A and 20 miles from station B, along the 100-mile parallel line from the shore.

Step 1: Understand the Problem Setup

The Coast Guard stations A and B are 200 miles apart on a straight shoreline. The ship emits a distress signal received at both stations. The ship is 100 miles from the shoreline, moving parallel to it.

Step 2: Interpret Given Information

The ship is 160 miles closer to station B than to station A. We need to find out the position of the ship relative to the two stations.

Step 3: Establish the Geometry

Consider the positions of the stations A and B as coordinates on a line where A is at (0,0) and B at (200,0). The ship is on a parallel line to the x-axis, 100 miles above it.

Step 4: Set Up the Equations

Let the position of the ship be (x, 100). Thus, the distance from A to the ship is \( \sqrt{x^2 + 100^2} \), and from B, it is \( \sqrt{(x-200)^2 + 100^2} \).

Step 5: Create the Equation from Conditions

Given that the ship is 160 miles closer to B than A, we set up the equation:\[\sqrt{x^2 + 100^2} - \sqrt{(x - 200)^2 + 100^2} = 160\]This reflects the distance difference.

Step 6: Solve the Equation

Simplify and solve the equation:1. Square both sides to eliminate the square roots: \[(\sqrt{x^2 + 100^2} - \sqrt{(x - 200)^2 + 100^2})^2 = 160^2\]2. Further expansion and simplification will eventually yield a solvable quadratic equation. Solving for x will provide the x-coordinate of the ship's position.

Step 7: Solve and Validate

Solve the quadratic equation derived from the previous step to find the x-coordinate of the ship. Ensure that the solution satisfies the original conditions, such as the distance between the stations and the relative closeness to each.

Key Concepts

Quadratic Equations

Quadratic equations appear frequently in algebra. They are particularly useful when modeling situations where relationships between variables lead to equations involving squares.
Here are some basics to remember about quadratic equations:

• A quadratic equation usually takes the form: \[ax^2 + bx + c = 0\]
• The solutions to quadratic equations, called roots, can be real or complex numbers.
• The well-known quadratic formula \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]helps us find the roots of the equation.

In coordinate geometry, the position and distance problems often translate into quadratic equations.
Here, when trying to locate the ship, setting up the problem led to a quadratic equation derived from the conditions of the exercise.

Distance Formula

The distance formula is a fundamental concept in coordinate geometry. It's used to determine the distance between two points in a plane. Understanding how it works is essential for solving many geometry problems.
The formula is expressed as: \[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two points.
• The formula is based on the Pythagorean theorem, which describes how to find the hypotenuse of a right triangle.

In the context of the ship problem, the distance formula helps determine the exact distance from stations A and B to the ship, given their coordinates.
Calculating these distances allowed us to set up an equation based on the ship being closer to one station.

Coordinate Geometry

Coordinate geometry, or analytic geometry, merges algebra techniques with geometry. This field is crucial for solving a variety of geometric problems using an analytical approach.
Some key points about coordinate geometry include:

• Points are defined using coordinates on a plane (e.g., \((x, y)\)).
• Geometrical shapes and distances can be expressed in terms of coordinate equations.
• It allows the use of algebraic methods to solve geometric problems.

In the exercise regarding the ship, coordinate geometry helps in setting up the problem by defining the positions of the ship and the coast stations as coordinates.
This makes it easier to apply the distance formula and derive a quadratic equation to find the ship's location efficiently.
Through such problems, we can appreciate how coordinate geometry serves as a bridge between algebra and geometry.