Question

1. a rectangular pool 6 meters by 8 meters is to be surrounded by a walkway of width x meters. this problem has to do with the rectangular region taken up by the pool and walkway.

(f) find the value of x which makes the walkway have an area equal to the area of the pool.
type your answer in the space below.
hint: you have already found the expression for the area of the entire region (the walkway and the pool), but you only need the area of the walkway without the pool area on one side. then, you can set up the equation area of walkway = area of pool

Explanation:

Step1: Find pool area

Pool is rectangle: \( \text{Area}_{\text{pool}} = 6 \times 8 = 48 \, \text{m}^2 \)

Step2: Find total area (pool + walkway)

Total length: \( 6 + 2x \), total width: \( 8 + 2x \)
\( \text{Area}_{\text{total}} = (6 + 2x)(8 + 2x) \)

Step3: Find walkway area

\( \text{Area}_{\text{walkway}} = \text{Area}_{\text{total}} - \text{Area}_{\text{pool}} \)
Substitute: \( (6 + 2x)(8 + 2x) - 48 \)
Expand: \( 48 + 12x + 16x + 4x^2 - 48 = 4x^2 + 28x \)

Step4: Set walkway area = pool area

\( 4x^2 + 28x = 48 \)
Divide by 4: \( x^2 + 7x - 12 = 0 \)? Wait, no—wait, pool area is 48, so:
Wait, correction: \( 4x^2 + 28x = 48 \) → \( x^2 + 7x - 12 = 0 \)? No, wait, total area minus pool is walkway. Wait, maybe I messed up total dimensions. Wait, the pool is 6m (length) and 8m (width)? Wait the diagram shows 6m as the base, 8m as the side? Wait no, the problem says "rectangular pool 6 meters by 8 meters". So length 6, width 8? Wait the diagram is a trapezoid? Wait no, maybe the diagram is misdrawn, or it's a rectangle with walkway. Wait, maybe the pool is 6m (length) and 8m (width), so total length with walkway is \( 6 + 2x \), total width \( 8 + 2x \). Then total area is \( (6 + 2x)(8 + 2x) \). Walkway area is total minus pool: \( (6 + 2x)(8 + 2x) - 48 \). Set equal to pool area (48):
\( (6 + 2x)(8 + 2x) - 48 = 48 \)
\( (6 + 2x)(8 + 2x) = 96 \)
Expand: \( 48 + 12x + 16x + 4x^2 = 96 \)
\( 4x^2 + 28x + 48 - 96 = 0 \)
\( 4x^2 + 28x - 48 = 0 \)
Divide by 4: \( x^2 + 7x - 12 = 0 \)? Wait, discriminant: \( 49 + 48 = 97 \), not a square. Wait, maybe the pool is 6m and 8m, but the diagram is a rectangle, so maybe I mixed up length and width. Wait, maybe the pool is 8m by 6m, but no. Wait, maybe the total area is (8 + 2x)(6 + 2x), pool area 48, walkway area 48. So:
\( (8 + 2x)(6 + 2x) - 48 = 48 \)
\( 48 + 16x + 12x + 4x^2 - 48 = 48 \)
\( 4x^2 + 28x - 48 = 0 \)
Divide by 4: \( x^2 + 7x - 12 = 0 \). Wait, that gives \( x = \frac{ -7 \pm \sqrt{49 + 48} }{2} = \frac{ -7 \pm \sqrt{97} }{2} \), which is not nice. So maybe the pool is 6m and 8m, but the diagram is a trapezoid? Wait no, the problem says "rectangular pool". Wait, maybe the pool is 6m (width) and 8m (length), so total width \( 6 + 2x \), total length \( 8 + 2x \). Then total area \( (8 + 2x)(6 + 2x) \), pool area 48, walkway area 48. So:
\( (8 + 2x)(6 + 2x) = 96 \)
\( 48 + 16x + 12x + 4x^2 = 96 \)
\( 4x^2 + 28x - 48 = 0 \)
Same as before. Wait, maybe the pool is 6m by 8m, but the walkway is around, so the total dimensions are (6 + 2x) and (8 + 2x), and walkway area is equal to pool area (48). So:
\( (6 + 2x)(8 + 2x) - 48 = 48 \)
\( (6 + 2x)(8 + 2x) = 96 \)
Let’s let \( y = x \), then \( (6 + 2y)(8 + 2y) = 96 \)
Expand: \( 48 + 12y + 16y + 4y^2 = 96 \)
\( 4y^2 + 28y - 48 = 0 \)
Divide by 4: \( y^2 + 7y - 12 = 0 \). Wait, discriminant is 49 + 48 = 97, which is not a perfect square. That can't be. So maybe I made a mistake in the pool dimensions. Wait, maybe the pool is 6m by 8m, but the walkway is only on two sides? No, the problem says "surrounded by a walkway", so all four sides. Wait, maybe the diagram is a trapezoid, so the pool is a trapezoid? Wait the diagram shows a trapezoid with bases 6m and (6 + 2x), and legs 8m? No, the problem says "rectangular pool". Maybe the diagram is incorrect, or the problem has a typo. Wait, maybe the pool is 8m by 6m, and the total area is (8 + 2x)(6 + 2x), pool area 48, walkway area 48. So:
Wait, maybe the pool is 6m (length) and 8m (width), so area 48. Walkway area equal to pool area means total area is 96. So (6 + 2x)(8 + 2x) = 96. Let's solve:
\( 4x^2 + 28x + 48 = 96 \)
\( 4x^2 + 28x - 48 = 0 \)
Divide by 4: \( x^2 + 7x - 12 = 0 \). Wait, that's not a nice answer. But maybe I messed up the pool dimensions. Wait, maybe the pool is 8m by 6m, but the problem says 6 meters by 8 meters. Wait, maybe the diagram is a rectangle, and the 8m is the length, 6m is the width. Wait, let's try x=1: total area (8)(10)=80, walkway area 80-48=32≠48. x=2: (10)(12)=120, walkway 72≠48. x= -12: no. Wait, maybe the pool is 6m by 8m, but the walkway is only on two sides? No, the problem says "surrounded". Wait, maybe the problem is a trapezoid? The diagram shows a trapezoid with top base (pool) and bottom base (pool + walkway), and legs 8m. Wait, trapezoid area: \( \text{Area} = \frac{(a + b)}{2} \times h \). Pool is a rectangle, so top base a=6, height h=8? No, the diagram has 8m as the side, 6m as the bottom base. Wait, maybe the pool is a trapezoid? No, the problem says rectangular. I think there's a mistake in my approach. Wait, let's re-express:

Wait, the problem says "the rectangular region taken up by the pool and walkway". So pool is rectangle, walkway surrounds it, so total is rectangle with length 6 + 2x, width 8 + 2x. Pool area: 6*8=48. Walkway area: (6+2x)(8+2x) - 48. Set equal to 48:

\( (6 + 2x)(8 + 2x) - 48 = 48 \)

\( (6 + 2x)(8 + 2x) = 96 \)

Expand:

\( 48 + 12x + 16x + 4x^2 = 96 \)

\( 4x^2 + 28x - 48 = 0 \)

Divide by 4:

\( x^2 + 7x - 12 = 0 \)

Using quadratic formula:

\( x = \frac{ -7 \pm \sqrt{49 + 48} }{2} = \frac{ -7 \pm \sqrt{97} }{2} \)

But \( \sqrt{97} \approx 9.85 \), so x ≈ ( -7 + 9.85 ) / 2 ≈ 1.425. But that's not a nice number. Maybe the pool is 8m by 6m, but the problem has a typo, or the diagram is a rectangle with length 8 and width 6, and walkway width x. Wait, maybe I mixed up length and width. Let's try length 8, width 6. Then total length 8 + 2x, total width 6 + 2x. Total area (8+2x)(6+2x). Pool area 48. Walkway area 48. So:

(8+2x)(6+2x) = 96

Same equation. So maybe the problem is correct, and the answer is \( \frac{ -7 + \sqrt{97} }{2} \approx 1.42 \), but that seems odd. Wait, maybe the pool is 6m by 8m, but the walkway is only on the length and width, not both sides? No, "surrounded" means all sides. Wait, maybe the diagram is a square? No. Wait, maybe the problem is from a source where the pool is 8m by 6m, and the walkway area is equal to pool area, so total area 96. Let's solve (8 + 2x)(6 + 2x) = 96:

Let’s expand: 48 + 16x + 12x + 4x² = 96 → 4x² + 28x - 48 = 0 → x² + 7x - 12 = 0. Using quadratic formula, x = [ -7 ± √(49 + 48) ] / 2 = [ -7 ± √97 ] / 2. Since x must be positive, x = ( -7 + √97 ) / 2 ≈ ( -7 + 9.849 ) / 2 ≈ 1.4245 meters. But maybe the problem has a typo, and the pool is 10m by 8m? No, the problem says 6 by 8. Alternatively, maybe the pool is 8m by 6m, and the walkway is only on two sides, but the problem says "surrounded". I think the intended problem is with a rectangle, and the equation is correct, so the answer is \( \frac{ -7 + \sqrt{97} }{2} \) or approximately 1.42. But maybe I made a mistake in the total area. Wait, let's check again:

Pool area: 6*8=48. Walkway area = pool area = 48. So total area = 48 + 48 = 96. So (6 + 2x)(8 + 2x) = 96. Yes, that's correct. So solving:

4x² + 28x + 48 = 96 → 4x² + 28x - 48 = 0 → x² + 7x - 12 = 0. Discriminant D = 49 + 48 = 97. So x = [ -7 + √97 ] / 2 ≈ 1.42.

Answer:

\( \frac{ -7 + \sqrt{97} }{2} \) (or approximately 1.42 meters)