Question

write an equation that relates the quantities. The perimeter \(p\) of a rectangle varies jointly with the sum of the lengths of its sides \(l\) and \(w\). The constant of proportionality is 2 .

Short answer

\( p = 2(l + w) \)

Step-by-step solution

Identify Given Variables

Given variables are:- Perimeter of the rectangle, denoted as \(p\)- Length of the rectangle, denoted as \(l\)- Width of the rectangle, denoted as \(w\)- Constant of proportionality, which is 2

Understand the Relationship

The problem states that the perimeter \(p\) varies jointly with the sum of the lengths of its sides, which means \(l + w\). This can be expressed as \(p \propto (l + w)\).

Include the Constant of Proportionality

To include the constant of proportionality in the equation, multiply the proportional relationship by 2 (the given constant). This gives us the equation: \[ p = 2 \times (l + w) \]

Write the Final Equation

Combine everything to write the final equation that relates the quantities:\[ p = 2(l + w) \]

Key concepts

Perimeter

In geometry, 'perimeter' refers to the total length around a two-dimensional shape. Specifically, for a rectangle, the perimeter is the sum of all its sides. Think of it like taking a walk around a park that has a rectangular shape; the distance you cover is the perimeter. It's important because it helps us understand the boundary length and can be applied in various real-life situations, such as fencing around a garden.

Rectangle

A rectangle is a four-sided polygon where opposite sides are equal in length, and each angle is a right angle (90 degrees). Rectangles are everywhere: books, screens, rooms, and more. Knowing the properties of a rectangle helps solve many practical problems. For example, if you know the length and width of a rectangle, you can easily determine its perimeter and area.

Proportionality

Proportionality describes how two quantities change together in a fixed ratio. In our exercise, the relationship between the perimeter and the sum of the length and width of the rectangle is described as a proportionality. The fact that the perimeter varies 'jointly' with the length and width means both these dimensions influence the perimeter together, maintaining a ratio defined by the given constant. Understanding this concept is key for solving various mathematical and real-world problems.

Equation

An 'equation' is a mathematical statement that asserts the equality of two expressions. In our exercise, we derived the equation for the perimeter of a rectangle using the concept of proportionality. The equation we wrote is:\[ p = 2(l + w) \]This equation tells us that the perimeter (\( p \)) is twice the sum of the length (\( l \)) and width (\( w \)) of the rectangle. It concisely captures the relationship among these quantities and allows us to calculate any of these variables if the others are known.