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Chapter 2: Problem 40

If the length of a rectangular parking lot is 10 meters less than twice its width, and the perimeter is 400 meters, find the length of the parking lot. (IMAGE CANNOT COPY)

Short Answer

Expert verified
The length of the parking lot is 130 meters.

Step by step solution

01

Define Variables

Let the width of the parking lot be \( x \) meters. Therefore, the length of the parking lot would be \( 2x - 10 \) meters as it is 10 meters less than twice the width.
02

Perimeter Equation

The perimeter of a rectangle is given by the formula \( 2 \times \text{length} + 2 \times \text{width} \). We know the perimeter is 400 meters, so we have:\[2(2x - 10) + 2x = 400\].
03

Simplify the Equation

Distribute and combine like terms in the perimeter equation:\[2 \times (2x - 10) + 2x = 400\]\[4x - 20 + 2x = 400\]Combine like terms:\[6x - 20 = 400\].
04

Solve for Width

Rearrange the equation to solve for \( x \):\[6x = 420\]\[x = 70\].Thus, the width of the parking lot is 70 meters.
05

Find the Length

Substitute the width back into the expression for the length:\[2x - 10 = 2(70) - 10 = 140 - 10 = 130\].The length of the parking lot is 130 meters.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rectangle Perimeter
The perimeter of a rectangle is the total distance around the edge of the rectangle. To calculate this, we need to add up the lengths of all four sides.
For a rectangle, we use the formula: \[ P = 2(l + w) \] where \( P \) is the perimeter, \( l \) is the length, and \( w \) is the width.This formula comes from the fact that a rectangle has two pairs of equal sides.
  • Two sides are the length \( l \).
  • Two sides are the width \( w \).
For example, if we know the perimeter is 400 meters, we can rearrange the formula to find missing values like length or width once one of these is known.
Algebraic Expressions
In mathematics, algebraic expressions are used to represent numbers and relationships between them using symbols and variables. In our exercise, the equation involves expressions with the width of the parking lot (\( x \)).
Algebraic expressions enable us to model real-world situations in a way that is flexible and easy to manipulate. For instance, if the length is described as "10 meters less than twice the width," we write this as \( 2x - 10 \).
  • \( 2x \) represents twice the width.
  • The "- 10" accounts for the length being less by 10 meters.
Using these expressions, we can plug them into equations, simplify, and solve to find unknown variables.
Problem-Solving Steps
Solving word problems involving equations requires a structured approach. Following problem-solving steps can make these problems easier to tackle and understand.
  • Define Variables: First, identify what the unknowns in the problem are and assign variables. In this exercise, the width is assigned as \( x \).
  • Write the Equation: Use any given conditions to set up an equation. The perimeter equation for a rectangle was used here.
  • Simplify: Combine like terms and simplify the equation to make it easier to solve.
  • Solve: Solve for the unknown variable. Often this involves basic arithmetic operations and manipulation.
  • Substitute Back: Once you've found a variable, substitute it back to find any other unknowns, such as the length in our exercise.
Breaking down the problem into these steps clarifies the process and ensures that each aspect is covered, leading to a complete solution.

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