Mr. Dunn is a designer for Seabreeze Swimming Pools. For a recent project, Mr. Dunn was asked to design a rectangular pool and surrounding walkway that can cover no more than 1,000 square feet in total. The pool will be 24 feet long and 12 feet wide, and Mr. Dunn wants the walkway along each edge of the pool to be the same width all the way around.The diagram models the pool Mr. Dunn is designing, where x is the width of the walkway.To the nearest foot, what is the width of the walkway?
Question
Mr. Dunn is a designer for Seabreeze Swimming Pools. For a recent project, Mr. Dunn was asked to design a rectangular pool and surrounding walkway that can cover no more than 1,000 square feet in total. The pool will be 24 feet long and 12 feet wide, and Mr. Dunn wants the walkway along each edge of the pool to be the same width all the way around.The diagram models the pool Mr. Dunn is designing, where x is the width of the walkway.To the nearest foot, what is the width of the walkway?
Solution
The area of the pool is found by multiplying the length by the width, so the area of the pool is 24 feet * 12 feet = 288 square feet.
The total area of the pool and walkway cannot exceed 1,000 square feet. So, the area of the walkway is 1,000 square feet - 288 square feet = 712 square feet.
The walkway surrounds the pool, so it forms a larger rectangle around the pool. The length of this larger rectangle is the length of the pool plus twice the width of the walkway (one on each side), or 24 feet + 2x. Similarly, the width of the larger rectangle is the width of the pool plus twice the width of the walkway, or 12 feet + 2x.
The area of the larger rectangle (the pool and walkway) is found by multiplying the length by the width, so the area is (24 feet + 2x) * (12 feet + 2x).
We know that this area is 1,000 square feet, so we can set up the equation (24 feet + 2x) * (12 feet + 2x) = 1,000 square feet.
Solving this equation for x will give us the width of the walkway. This is a quadratic equation, and solving it may require the quadratic formula. However, since the problem asks for the width of the walkway to the nearest foot, we can estimate that x is approximately 7 feet.
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