Problem 32 - College Algebra with Modeling and Visualization

Question

Two ships leave a harbor at the same time. The first ship heads north at 20 miles per hour, and the second ship heads west at 15 miles per hour. Write an expression that gives the distance \(d\) between the ships after \(t\) hours.

Step-by-Step Solution

Answer

The distance \(d\) is \(25t\) miles.

Step 1: Understand the Problem

We need to find the expression for the distance between two ships moving in perpendicular directions: one goes north and the other west. After some time \(t\), the ships form a right triangle with the harbor at the origin.

Step 2: Recall the Pythagorean Theorem

The distance between the ships corresponds to the hypotenuse of a right triangle, where the legs are the distances traveled by each ship. According to the Pythagorean theorem for a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\), we have \(c^2 = a^2 + b^2\).

Step 3: Calculate Each Ship's Distance Traveled

The first ship, traveling north at 20 mph, covers \(20t\) miles in \(t\) hours. The second ship, traveling west at 15 mph, covers \(15t\) miles in \(t\) hours.

Step 4: Apply the Pythagorean Theorem

Substitute the ship distances into the Pythagorean theorem: \[ d^2 = (20t)^2 + (15t)^2 \].

Step 5: Simplify the Expression

Simplify the terms: \(d^2 = 400t^2 + 225t^2 = 625t^2\). Take the square root of both sides to solve for \(d\): \(d = \sqrt{625t^2}\).

Step 6: Simplify the Square Root

Simplify \(\sqrt{625t^2}\) to get \(d = 25t\), since \(\sqrt{625} = 25\).

Key Concepts

Distance Formula

The distance formula is a fundamental concept in geometry and plays a crucial role in determining the distance between two points in a plane. It is derived from the Pythagorean Theorem.
To calculate the distance between two points with coordinates \(x_1, y_1\) and \(x_2, y_2\), the formula used is:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]In the context of the problem with the two ships, since they move in perpendicular directions, we don't need to directly consider coordinates. Instead, their movements form a right triangle. The ships' paths are the perpendicular legs of this triangle.
This geometric understanding is what allows us to apply the Pythagorean theorem, as their combined paths and the port of departure form a right triangle. Calculating the distance between them involves finding the hypotenuse of this triangle, which is why the formula becomes \(d = 25t\) after simplifying. Thus, after \(t\) hours, the distance between the ships is \25t\ miles.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations that describe a particular relationship or value. In the exercise, we form an algebraic expression to describe the distance between two ships using the variable \(t\), which represents time.

• The northern ship travels at 20 mph, covering \(20t\) miles in \(t\) hours.
• The western ship travels at 15 mph, covering \(15t\) miles in \(t\) hours.

Each of these expressions represents a ship's journey over time. To find the overall distance between the ships, these expressions must be inserted into the Pythagorean equation: \(d^2 = (20t)^2 + (15t)^2\).
Next, we simplify it to \(d^2 = 400t^2 + 225t^2\), which combines to form \(d^2 = 625t^2\). Finally, taking the square root provides \(d = 25t\), a clean and simple expression for the distance, making use of algebraic manipulation.

Right Triangle

A right triangle is a triangle in which one of the angles is exactly 90 degrees. Right triangles are integral to many mathematical concepts due to the reliable relationship between their sides and angles.
The exercise involving the ships forms a natural right triangle, with the harbor as the right angle, one ship heading north, and the other west. These paths are the "legs" of the triangle.

• The north path is a vertical leg.
• The west path is a horizontal leg.

The distance between the ships, \(d\), forms the "hypotenuse."
This configuration allows us to apply the Pythagorean Theorem — stating that in a right triangle, the square of the hypotenuse's length equals the sum of the squares of the two legs. This theorem is the foundation for solving this problem, as it created a straightforward way to calculate the distance, \(d = 25t\), after simplifying the expressions.