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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?

Respuesta :

Space

Answer:

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

Step-by-step 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.

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

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