Question

samantha is planning to install artificial grass in her backyard with a maximum area of 180 square feet allowed by her homeowner’s association. she wants a rectangular area of turf with a length 2 feet more than twice the width. find the length and width.\
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a. width: 20 feet, length: 9 feet\
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b. width: 9 feet, length: 20 feet\
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c. width: 6 feet, length: 30 feet\
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d. width: 11 feet, length: 24 feet

Explanation:

Step1: Define variables for width/length

Let width = $w$. Length $l = 2w + 2$.

Step2: Set up area equation

Area $A = l \times w = 180$, so:
$$(2w + 2)w = 180$$

Step3: Simplify quadratic equation

Expand and rearrange:
$$2w^2 + 2w - 180 = 0$$
Divide by 2:
$$w^2 + w - 90 = 0$$

Step4: Factor quadratic equation

Factor to solve for $w$:
$$(w + 10)(w - 9) = 0$$
Solutions: $w = 9$ (positive value only, since width can't be negative)

Step5: Calculate length

Substitute $w=9$ into $l=2w+2$:
$$l = 2(9) + 2 = 20$$

Step6: Verify area

Check $9 \times 20 = 180$, which matches the maximum allowed area.

Answer:

B. width: 9 feet, length: 20 feet