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Chapter 1: Problem 34

Solve each problem. Use all five problem-solving steps. The width of a rectangular garden is one-third of its length, and its perimeter is \(32 \mathrm{m} .\) Find the dimensions of the garden.

Short Answer

Expert verified
Length: 12 meters, Width: 4 meters

Step by step solution

01

- Understand the Problem

The problem states that the width of the garden is one-third of its length, and the perimeter is 32 meters. We are to find the length and width of the garden.
02

- Devise a Plan

Let's denote the length of the garden as L meters and the width as W meters. According to the problem, W = (1/3)L. The formula for the perimeter of a rectangle is given by P = 2L + 2W. We will substitute the given relationships into this formula.
03

- Carry Out the Plan

Using the given perimeter, we have the equation: \[ 2L + 2W = 32 \] Substitute W = (1/3)L into the perimeter equation: \[ 2L + 2 \left( \frac{L}{3} \right) = 32 \] Combine the terms: \[ 2L + \frac{2L}{3} = 32 \] To eliminate the fraction, multiply every term by 3: \[ 6L + 2L = 96 \] Combine like terms: \[ 8L = 96 \] Divide both sides by 8: \[ L = 12 \]
04

- Find the Width

Now, using the relationship between the width and length \[ W = \frac{L}{3} \] Substitute L = 12: \[ W = \frac{12}{3} = 4 \]
05

- Review and Check

Let's check the dimensions to ensure they satisfy the perimeter condition. Length: 12 meters, Width: 4 meters. Calculate the perimeter: \[ 2L + 2W = 2(12) + 2(4) = 24 + 8 = 32 \] The calculated perimeter is 32 meters, which matches the given condition. Thus, the dimensions are correct.

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Key Concepts

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

Problem-Solving Steps
Solving problems step-by-step is essential to finding a clear and accurate solution. In this rectangular garden problem, we use five key steps to guide us: Understand the Problem, Devise a Plan, Carry Out the Plan, Find the Width, and Review and Check. By breaking down the problem, we gradually solve it without missing details.
Rectangular Garden Dimensions
Understanding the dimensions of a rectangular garden is crucial. In the given problem, we learn that the width is one-third of the length. We denote the length by L meters and the width by W meters. Therefore, W = (1/3)L. By knowing how the length and width relate to each other, we can use this information to solve for both dimensions.
Perimeter Formula
The perimeter formula for a rectangle is essential for solving this problem. The formula is P = 2L + 2W where P is the perimeter, L is the length, and W is the width. Given that the perimeter is 32 meters, we substitute L and W into the equation to form 2L + 2W = 32. This equation, along with the relationship W = (1/3)L, allows us to solve for L and then W. Substituting W into the perimeter formula and simplifying the equation leads us to the solution.

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Most popular questions from this chapter

Which numbers in the list provided are (a) whole numbers? (b) integers? (c) rational numbers? (d) irrational numbers? (e) real numbers?. $$ -\frac{9}{2},-4,-1.2,0, \sqrt{5}, 3 $$

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