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Difficulty: MediumPerimeter and Area of 2D Figures

A rectangular community garden is 33 times as long as it is wide. If the total area of the garden is 108108 square feet, what is the perimeter of the garden, in feet?

  1. 4848Answer
  2. B
    108108
  3. C
    4242
  4. D
    2424

Answer

The perimeter of the garden is 4848 feet.
The correct answer is found by first determining the dimensions of the garden. Since the length is 33 times the width, the area can be represented as (3w)(w)=3w2(3w)(w) = 3w^2. Setting this equal to the given area of 108108 square feet gives 3w2=1083w^2 = 108. Dividing by 33 yields w2=36w^2 = 36, so the width is 66 feet. The length is 3×6=183 \times 6 = 18 feet. The perimeter formula for a rectangle is P=2(length+width)P = 2(length + width). Substituting the dimensions gives P=2(18+6)=2(24)=48P = 2(18 + 6) = 2(24) = 48 feet.

Step-by-Step Solution

1
Let the width of the garden be ww. The length is 33 times the width, so the length is 3w3w.
Width = ww, Length = 3w3w
Translating the word problem into algebraic expressions allows us to set up an equation.
2
Set up an equation using the formula for the area of a rectangle (Area=length×widthArea = length \times width).
3w×w=1083w \times w = 108
We are given the total area, which links our variable expressions to a known value.
3
Solve the equation for ww.
3w2=108⇒w2=36⇒w=63w^2 = 108 \Rightarrow w^2 = 36 \Rightarrow w = 6
Isolating the variable determines the numerical value for the width.
4
Substitute w=6w = 6 back into the expression for length (3w3w).
Length = 3(6)=183(6) = 18 feet
Both dimensions are needed to calculate the perimeter.
5
Calculate the perimeter using the formula P=2(length+width)P = 2(length + width).
P=2(18+6)=2(24)=48P = 2(18 + 6) = 2(24) = 48 feet
The perimeter represents the total distance around the outside of the rectangular garden.

Key Concept

Perimeter and Area of 2D Figures
Estimated Time:1m 0s
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Topics

Perimeter and Area of 2D Figures

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