Problem 61 - Algebra A Combined Function

Question

Solve the following. For Exercises 61 and \(62,\) the solutions have been started for you. The perimeter of a rectangle is to be no greater than 100 centimeters and the width must be 15 centimeters. Find the maximum length of the rectangle.

Step-by-Step Solution

Answer

The maximum length of the rectangle is 35 centimeters.

Step 1: Understand the Problem

The problem requires us to find the maximum possible length of a rectangle given that the perimeter cannot exceed 100 centimeters and the width is fixed at 15 centimeters.

Step 2: Formulate the Perimeter Equation

The perimeter \(P\) of a rectangle is given by the formula \(P = 2(l + w)\), where \(l\) is the length and \(w\) is the width. For our problem, this equation must be no greater than 100 cm. Since the width is 15 cm, the equation becomes \(2(l + 15) \leq 100\).

Step 3: Simplify the Inequality

Simplify the inequality \(2(l + 15) \leq 100\). First, divide both sides by 2: \[l + 15 \leq 50\].

Step 4: Solve for Length

Subtract 15 from both sides of the inequality to solve for \(l\):\[l \leq 35\]. This tells us that the maximum possible length is 35 centimeters.

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values, where they might not be exactly equal but have a greater or lesser relationship. Understanding inequalities is essential because it helps us determine the range of values a variable can take.
For example, in the context of the rectangle perimeter problem, we use the inequality to express that the perimeter can't be more than 100 centimeters. This is written as \(2(l + 15) \leq 100\). It implies the perimeter must be less than or equal to 100.
Inequalities can be solved by:

• Adding or subtracting the same number from both sides.
• Multiplying or dividing both sides by a positive number.
• Being cautious with negative numbers: when you multiply or divide by a negative, the inequality flips direction.

Understanding these steps helps you manipulate the inequality to find the required solution, like determining the maximum length of the rectangle.

Rectangle Geometry

Rectangle geometry deals with the properties and characteristics of rectangles.
A rectangle is a four-sided figure with opposite sides equal and all angles at 90 degrees. It has two essential parameters:

• Length (l) - the longer side of the rectangle.
• Width (w) - the shorter side of the rectangle.

The formula for the perimeter (P) of a rectangle expresses the total distance around the figure and is calculated as \(P = 2(l + w)\). Here, the perimeter is determined by adding the length and width, then multiplying by two since each side appears twice.
Understanding rectangle geometry helps represent and solve practical problems like our perimeter example, where you can use dimensions to find how large each side can be while adhering to given constraints.

Problem Solving

Problem-solving in mathematics involves several steps, from understanding a problem to reaching a solution.
Here's a simple process:

• Understand the Problem: What do we need to find or solve?
• Plan: Figure out what information and formulas can help solve the problem.
• Execute: Implement the plan by substituting known values into formulas and solving equations or inequalities.
• Review: Check your solution to ensure it makes sense and meets the problem requirements.

In the rectangle perimeter problem, we started by understanding that we need the maximum possible length. We utilized the perimeter formula and constraints to construct an inequality and solved it to find a satisfactory length.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a value. They are crucial in forming equations that need solving.
In our problem, we began with the algebraic expression for the perimeter of a rectangle: \(P = 2(l + w)\). Here, P is the perimeter, l is the length, and w is the width.
By substituting known values like width \(w = 15\), we simplify to find the relationship between length and perimeter. This transforms the expression into an inequality \(2(l + 15) \leq 100\), which we further simplify and solve.
Using algebraic expressions in geometry allows us to translate real-world problems into solvable mathematical formats, guiding us to answers such as the maximum allowable measurement of one side of the rectangle.