Question

Is it possible for a rectangle with a perimeter of 52 centimeters to have an area of 148.75 square centimeters? Explain.

Short answer

So, yes, it is possible for a rectangle with a perimeter of 52 centimeters to have an area of 148.75 square centimeters, by having a length of 12.5 centimeters and width of 13.5 centimeters.

Step-by-step solution

Calculate the length and width from the perimeter

First, set up the equation for the perimeter of a rectangle, which is \(2 \times (length + width) = 52\). Simplifying this gives \(length + width = 26\). This represents all the combinations of length and width that total 26.

Set up the equation for the area

The area of a rectangle is given by the multiplication of its length and width. Denoting the length as \(l\) and the width as \(w\), the area of the rectangle can be formulated as \(l\times w = 148.75\).

Verify if given area is possible

To verify if the given area can be achieved with the given perimeter, it can be checked if there are values of length and width that satisfy both equation from step 1 and 2 simultaneously. Given that they are simultaneous equations, the most straightforward method would be substitution or elimination method. However, considering that this is a quadratic equation, it's better to rewrite it as in the form \(x^2 -sx +p = 0\), where \(s\) is the sum (perimeter/2) and \(p\) is the product (area), we have \(x^2 -26x + 148.75 = 0\). By solving this, we get two valid roots 12.5 and 13.5, implying that a length of 12.5 cm and width of 13.5 cm can indeed give the required area and perimeter.

Key concepts

Perimeter of a Rectangle

Understanding the perimeter of a rectangle is essential when dealing with various geometric problems. In essence, the perimeter is the total distance around the outside of the rectangle. It is the sum of all its sides. To calculate it, you can simply double the sum of the length and width of the rectangle. The formula can be written as: \[ P = 2(l + w) \] where \( P \) stands for the perimeter, \( l \) for length, and \( w \) for width.

This concept is crucial when you need to find the dimensions of a rectangle when given the perimeter. If the problem includes the perimeter, as in our exercise, you can start by setting up an equation as shown in the first step of our solution. Completing this step provides a foundation for working with other properties of the rectangle, such as its area.

Area of a Rectangle

While the perimeter is about the boundary, the area of a rectangle refers to the amount of space enclosed within it. Simply put, it's a measure of the surface the rectangle covers. Calculating the area is straightforward when you know its length and width; you multiply these two dimensions. The formula for the area is: \[ A = l \times w \] where \( A \) represents the area.

In the exercise, step 2 demonstrates the use of this concept perfectly. Knowing the area allows us to link it with the perimeter, thus setting up the stage for solving simultaneous equations—a process critical for determining if a rectangle with specific dimensions can exist, given constraints on the area and perimeter.

Simultaneous Equations

Simultaneous equations are a set of equations with multiple unknowns that are true at the same time. They can be solved to find the values of these unknowns. There are different methods to solve simultaneous equations, such as substitution, elimination, and graphical methods.

In our rectangle problem, we are dealing with two unknowns, length and width, which must satisfy both the perimeter and area equations simultaneously. The task is to find a common solution that fits both equations. Solving simultaneous equations provides the link between various characteristics of geometric shapes, such as their perimeter and area in this case.

Quadratic Equations

Quadratic equations are an integral part of algebra, and they come into play when dealing with problems involving areas and dimensions, among other topics. A standard quadratic equation is in the form of \[ ax^2 + bx + c = 0 \] with \( a \), \( b \), and \( c \) being constants and \( x \) representing the unknown variable.

As shown in the exercise's step 3, we can transform the equations for the perimeter and area into a quadratic equation when the expressions are set to equal each other. By applying the quadratic formula or factoring, we can obtain the dimensions of the rectangle. When the problem involves a rectangle's perimeter and area like ours, the roots of the quadratic equation will reveal the potential dimensions that satisfy both conditions.