Question

Janice puts a fence around her rectangular garden. The garden has a length that is 9 feet less than 3 times its width. What is the perimeter of Janice's fence if the area of her garden s 5,670 square feet?

Answer 1

Let width be x ft

Length be 3x-9ft

Now

• Area=Length ×Breadth
• 5670=x(3x-9)
• 5670=3x²-9x
• 3x²-9x-5670=0

On solving we wI'll get x=-42,45

Take it positive

• x=45
• width=45ft
• Length=3(45)-9=135-9=126ft

Now

Perimeter

• 2(Length+Width)
• 2(45+126)
• 2(171)
• 342ft

Answer 2

P = 342 ft

General Formulas and Concepts:

Pre-Algebra

Evaluations

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Distributive Property

Algebra I

Equality Properties

• Multiplication Property of Equality
• Division Property of Equality
• Addition Property of Equality
• Subtraction Property of Equality

Terms/Coefficients

• Factoring/Expanding

Quadratics

• Solving quadratic equations
• Standard Form: ax² + bx + c = 0

Geometry

Area of a Rectangle Formula: A = lw

• l is length
• w is width

Perimeter of a Rectangle Formula: P = 2w + 2l

• l is length
• w is width

Explanation:

Step 1: Define

Identify variables.

A = 5670 ft²

"Length that is 9 feet less than 3 times its width" → l = 3w - 9

Step 2: Find w

1. Substitute in variables [Area of a Rectangle Formula]: w(3w - 9) = 5670
2. [Distributive Property] Expand: 3w² - 9w = 5670
3. [Subtraction Property of Equality] Rewrite [Standard Form]: 3w² - 9w - 5670 = 0
4. [Quadratic] Factor: 3(w - 45)(w + 42) = 0
5. [Quadratic] Solve [Find roots]: w = -42, 45

Since length/width cannot be negative, we know that w = 45 ft.

Step 3: Find l

1. Substitute in w [Length Formula]: l = 3(45) - 9
2. [Order of Operations] Evaluate: l = 126 ft

Step 4: Find Perimeter

1. Substitute in variables [Perimeter of a Rectangle Formula]: P = 2(45 ft) + 2(126 ft)
2. [Order of Operations] Evaluate: P = 342 ft

∴ The perimeter of Janice's fence is 342 ft.