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Question

Difficulty: MediumOperations on Polynomials

A rectangular swimming pool has a length of 3x−23x - 2 meters and a width of 2x−52x - 5 meters. A concrete walkway with a uniform width of 22 meters surrounds the pool. Which of the following expressions represents the area, in square meters, of the walkway?

  1. 20x−1220x - 12Answer
  2. B
    −18x+8-18x + 8
  3. C
    20x+820x + 8
  4. D
    12x2−18x+812x^2 - 18x + 8
  5. E
    12x2−1212x^2 - 12

Answer

The expression representing the area of the walkway is 20x−1220x - 12.
The correct answer is the expression 20x−1220x - 12. The total area including the walkway is (3x+2)(2x−1)=6x2+x−2(3x + 2)(2x - 1) = 6x^2 + x - 2. The pool's area is (3x−2)(2x−5)=6x2−19x+10(3x - 2)(2x - 5) = 6x^2 - 19x + 10. Subtracting the pool's area from the total area and distributing the negative sign to all terms gives: (6x2+x−2)−(6x2−19x+10)=6x2+x−2−6x2+19x−10=20x−12(6x^2 + x - 2) - (6x^2 - 19x + 10) = 6x^2 + x - 2 - 6x^2 + 19x - 10 = 20x - 12.

Step-by-Step Solution

1
Determine the dimensions of the entire area including the pool and the walkway.
The total length is (3x−2)+2(2)=3x+2(3x - 2) + 2(2) = 3x + 2 meters, and the total width is (2x−5)+2(2)=2x−1(2x - 5) + 2(2) = 2x - 1 meters.
Since the walkway surrounds the pool on all sides, we add twice the width of the walkway to both the length and width of the pool.
2
Calculate the total area including the walkway.
Total Area = (3x+2)(2x−1)=6x2−3x+4x−2=6x2+x−2(3x + 2)(2x - 1) = 6x^2 - 3x + 4x - 2 = 6x^2 + x - 2 square meters.
The area is found by multiplying the total length and the total width using binomial multiplication.
3
Calculate the area of the pool itself.
Pool Area = (3x−2)(2x−5)=6x2−15x−4x+10=6x2−19x+10(3x - 2)(2x - 5) = 6x^2 - 15x - 4x + 10 = 6x^2 - 19x + 10 square meters.
The area of the pool is the product of its length and width.
4
Subtract the pool area from the total area to find the area of the walkway.
Walkway Area = (6x2+x−2)−(6x2−19x+10)=6x2+x−2−6x2+19x−10=20x−12(6x^2 + x - 2) - (6x^2 - 19x + 10) = 6x^2 + x - 2 - 6x^2 + 19x - 10 = 20x - 12 square meters.
Subtracting the inner pool area from the total area yields the remaining area of the walkway surrounding it.

Key Concept

Calculating the difference between two binomial products to find the area of a border.
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Topics

Operations on Polynomials

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