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Difficulty: HardPerimeter and Area of 2D Figures

A rectangular swimming pool measures 4040 feet in length and 3030 feet in width. A concrete deck of uniform width is built around the entire outer edge of the pool. If the area of the concrete deck alone is 296296 square feet, what is the width of the deck, in feet?

Answer: 2 feet

Answer

2
The correct width is found by setting up an equation for the total area. The deck adds its width to both sides of the pool's length and width, making the overall dimensions 40+2w40 + 2w and 30+2w30 + 2w. The area of the deck alone is the total area minus the pool's area. Solving the algebraic equation (40+2w)(30+2w)−1200=296(40 + 2w)(30 + 2w) - 1200 = 296 yields a positive root of w=2w = 2.

Step-by-Step Solution

1
Calculate the area of the pool.
40×30=120040 \times 30 = 1200 square feet.
You need the baseline area of the inner rectangle before the deck is added.
2
Write algebraic expressions for the outer dimensions of the pool and deck combined.
Length =40+2w= 40 + 2w, Width =30+2w= 30 + 2w.
The deck has a uniform width ww, which adds to both the top/bottom and left/right sides of the pool.
3
Formulate an equation for the area of the deck.
(40+2w)(30+2w)−1200=296(40 + 2w)(30 + 2w) - 1200 = 296.
The area of the deck alone is equal to the total area (pool plus deck) minus the area of the pool.
4
Expand the binomials and simplify into a standard quadratic equation.
1200+140w+4w2−1200=296  ⟹  4w2+140w−296=01200 + 140w + 4w^2 - 1200 = 296 \implies 4w^2 + 140w - 296 = 0.
Standard quadratic form (ax2+bx+c=0ax^2 + bx + c = 0) is required to solve for the unknown variable ww.
5
Solve the quadratic equation for ww.
Divide by 44 to get w2+35w−74=0w^2 + 35w - 74 = 0. Factoring yields (w+37)(w−2)=0(w + 37)(w - 2) = 0, so w=2w = 2.
Finding the positive root gives the actual physical width of the concrete deck.

Key Concept

Calculating the area of composite rectangular figures and solving quadratic equations derived from expanding geometric boundaries.
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Topics

Perimeter and Area of 2D Figures

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