Problem 2 - College Algebra with Modeling and Visualization

Question

Evaluate the function for the indicated inputs and interpret the result. \(A(20,35),\) where \(A(w, l)=w l(A\) computes the area of a rectangle with width \(w\) and length \(L\) )

Step-by-Step Solution

Answer

The area of the rectangle is 700 square units.

Step 1: Understand the Function

The function given is \(A(w, l) = w \times l\), which represents the area of a rectangle. In this function, \(w\) stands for the width of the rectangle, and \(l\) represents the length. The output of the function provides the area by multiplying width and length.

Step 2: Substitute the Values

Substitute the given values for width and length into the function: \(w = 20\) and \(l = 35\). This means we need to calculate \(A(20, 35) = 20 \times 35\).

Step 3: Calculate the Area

Perform the multiplication to find the area: \[A(20, 35) = 20 \times 35 = 700.\] This product gives us the area of the rectangle with the specified width and length.

Step 4: Interpret the Result

The result of 700 represents the area of the rectangle in square units. This area is derived from a rectangle that is 20 units wide and 35 units long, meaning there are 700 square units within this rectangle.

Key Concepts

Rectangle Area

Understanding the area of a rectangle is crucial in geometry, as it forms the foundation for many other concepts. A rectangle is a four-sided shape with opposite sides that are equal in length. The area of a rectangle refers to the amount of space within the boundary of this shape. It is measured in square units, such as square inches or square centimeters, depending on the units used for length and width.

To calculate a rectangle's area, you'll need to know two key measurements:

• The width (sometimes referred to as the breadth or height in different contexts).
• The length (also known as the base or depth, depending on the situation).

The formula for finding the area is simple: Multiply the width by the length. This gives the total number of unit squares that can fit inside the rectangle, effectively measuring its size.

Mathematical Function

In mathematics, a function represents a relationship between a set of inputs and a set of possible outputs. The core idea is that each input is associated with precisely one output. In our case, the function provided is \(A(w, l) = w \times l\), which indicates that for every pair of width and length values, there is one distinct rectangle area.

A function like this is called a mathematical or algebraic function because it involves algebraic operations (here, multiplication). The inputs \(w\) and \(l\) are variables, which means they can take on different values. The multiplication operation applied to these variables provides the calculated output, which is the area. Functions are indispensable tools in mathematics for modeling real-world situations and solving problems systematically. They help us predict outcomes based on certain variables or conditions.

Multiplication of Dimensions

Multiplication of dimensions is the process of calculating the product of two length measurements, which collectively determine the size of a two-dimensional space. In the context of rectangle area calculation, the dimensions are width and length. By multiplying these dimensions, we efficiently find the total area.

Let's break down the calculation done in the example to solidify our understanding. For instance, the computation for \(A(20, 35)\) implies:

• Taking the width (20 units) and
• Multiplying it by the length (35 units).

The multiplication works as follows:\[20 \times 35 = 700\],resulting in the rectangle's area being 700 square units. Here, multiplication serves as a shortcut to adding each unit of width across every unit of length, which would otherwise be a lengthy and impractical method. This kind of multiplication is essential for computing areas, volumes, and other spatial measurements across different fields like architecture, engineering, and design.