Problem 104 - Prealgebra

Question

A dozen centuries-old shipwrecks were found in the Baltic Sea by a gas company building an underwater pipeline between Russia and Germany. The 12 wrecks were found in a 30-mile-long and \(1.2\) -mile-wide corridor at a depth of 430 feet. Model the corridor with a rectangle and find the approximate area of the region where the ships were found. Associated Press-Times-Standard 03/10/10 Centuries-old shipwrecks found in Baltic Sea.

Step-by-Step Solution

Answer

The area of the region is 36 square miles.

Step 1: Understand the Problem

We are tasked with finding the area of a rectangular region where shipwrecks were found. The dimensions provided are a length of 30 miles and a width of 1.2 miles.

Step 2: Identify the Formula for Area

The area of a rectangle is calculated by the formula \( \text{Area} = \text{Length} \times \text{Width} \). We'll use this formula to find the area of the region.

Step 3: Substitute the Values

Substitute the given values into the area formula: \( \text{Area} = 30 \text{ miles} \times 1.2 \text{ miles} \).

Step 4: Calculate the Area

Perform the multiplication: \( 30 \times 1.2 = 36 \). Thus, the area is 36 square miles.

Key Concepts

Area of a Rectangle

When we talk about the area of a rectangle, we refer to the space inside the rectangle's boundaries. Calculating the area is essential in many real-life applications. For example, in the exercise at hand, we need to determine the size of an area to understand where shipwrecks were found in the Baltic Sea. To find the area, use the simple formula:

• \( \text{Area} = \text{Length} \times \text{Width} \)

This equation tells us that by multiplying the length and the width of the rectangle, we get the measure of the total space inside the shape. The units of the area will be in "square units," based on the units used for length and width. In our case, since both dimensions are in miles, the area will be in square miles.

Mathematical Modeling

Mathematical modeling helps us represent real-world scenarios using mathematical concepts. In this exercise, we use a rectangle to model the corridor where the shipwrecks were discovered. This model simplifies the real world into manageable mathematical parameters. Why choose a rectangle? Because rectangles can effectively represent regions that have defined boundaries. Rectangles are easy to work with mathematically.

• They're defined by just two dimensions: length and width.
• Their area can be simply calculated using a multiplication formula.

Mathematical models are powerful. They allow us to estimate and calculate quantities even when dealing with spaces that might be much more complex in reality.

Problem Solving

Solving problems in math means breaking down the issue into manageable steps. For this exercise, we needed to find the area of a rectangular region:

• Start by understanding what is asked: finding the area where shipwrecks were found.
• Identify the right formula: use the area formula for a rectangle.
• Substitute the known values into the formula: given length and width.
• Perform the calculations to find the solution: multiply the length by the width to get the area.

These steps illustrate a structured method not just for this specific exercise, but are applicable to any similar problems. By following this approach, students can solve complex problems effectively, ensuring every step logically follows the previous one.