Problem 44 - Discrete Mathematics with Applications

Question

Solve each equation. $$P(5, r)=20$$

Step-by-Step Solution

Answer

The width (r) of the rectangle is 5, because we are given the perimeter (P) of 20 and the length (l) of 5. Using the formula for the perimeter of a rectangle \(P = 2l + 2w\), we simply plug in the given values and solve for r resulting in \(r=5\).

Step 1: Write down the given information and the formula for the perimeter of a rectangle

We are given the following information: - Length (l) = 5 - Perimeter (P) = 20 We need to find the width (r) of the rectangle. The formula for the perimeter of a rectangle is given by: $$ P = 2l + 2w $$

Step 2: Substitute the given values into the formula

Replace \(P\) with 20 and \(l\) with 5 in the formula: $$ 20 = 2(5) + 2r $$

Step 3: Simplify and solve for r

First, simplify the equation by multiplying on the left side: $$ 20 = 10 + 2r $$ Now, subtract 10 from both sides to isolate r: $$ 10 = 2r $$ Finally, divide both sides by 2: $$ r = 5 $$

Step 4: Write down the final answer

The width (r) of the rectangle is 5.

Key Concepts

Equation Solving

When we talk about solving equations, we're figuring out the value of an unknown variable that makes the equation true. In the context of geometry, many problems involve finding dimensions of shapes, like rectangles, based on given criteria.

Here, we were given the perimeter of a rectangle and asked to find its width. Equation solving involves substituting known values into a formula and isolating the unknown.

• First, you rewrite the general formula specific to your problem using known values.
• Next, simplify the equation step-by-step: combine like terms, and use inverse operations to isolate the variable of interest.
• In this problem, subtraction and division were used to solve for the width of the rectangle after simplifying the formula.

Solving equations requires a strategic approach, patiently handling each step until the variable is isolated, leading to the correct solution.

Rectangle Dimensions

Understanding the dimensions of a rectangle is key to solving many geometric questions. A rectangle is defined by two dimensions: length and width. Knowing these dimensions allows us to calculate various properties, such as the perimeter and area.

In this exercise, one side, the length, was provided, and the perimeter was given. The task was to deduce the missing dimension—the width. Knowing the relationship between these dimensions allows for solving complex geometric problems by using formulas correctly.

A rectangle is very versatile as calculating one missing value often relies on other properties, integrating other concepts like using diagonals or symmetry. Understanding how dimensions relate to one another in a geometric shape is foundational to solving such exercises confidently.

Geometry Formulas

Formulas in geometry are essential tools used to calculate various properties of shapes. The formula for the perimeter of a rectangle is \[ P = 2l + 2w \] which sums twice the length and twice the width, circumscribing the entire shape.

Geometry formulas serve as a bridge from known quantities to unknowns, aiding in the derivation of one dimension from another given some primary information. They simplify complex real-world measurements into manageable, solvable equations.

• Formulas are universal for specified geometric shapes and can be applied consistently across different problems.
• They represent the theoretical basis for geometric reasoning and problem-solving.
• Knowing these formulas by heart is crucial as they are frequently used in various practical applications such as engineering, architecture, and design.

Acquiring mastery of these formulas is essential for anyone embarked on learning geometry, forming a foundation for deeper exploration into the subject.