Question

1. think about a plan you have a rectangular koi pond that measures 6 ft by 8 ft. you have enough concrete to cover 72 ft² for a walkway, as shown in the diagram. what should the width of the walkway be? how can you write the outer dimensions of the walkway? how can you represent the total area of the walkway and pond in two ways?

Explanation:

Step1: Define width variable

Let width = \( w \) ft
Outer dimensions: \( 6 + 2w \), \( 8 + 2w \)

Step2: Total area equation

Total area = pond + walkway = \( 6×8 + 72 = 120 \)
Also total area = \( (6 + 2w)(8 + 2w) \)

Step3: Expand and simplify

\( (6 + 2w)(8 + 2w) = 120 \)
\( 48 + 28w + 4w^2 = 120 \)
\( 4w^2 + 28w - 72 = 0 \)
\( w^2 + 7w - 18 = 0 \)

Step4: Solve quadratic equation

\( (w + 9)(w - 2) = 0 \)
Valid solution: \( w = 2 \)

Step5: Outer dimensions and total area

Outer dimensions: \( 6 + 2×2 = 10 \), \( 8 + 2×2 = 12 \)
Total area representations: \( (6 + 2w)(8 + 2w) \) and \( 48 + 72 \)

Answer:

2 ft
(6 + 2w) ft by (8 + 2w) ft

1. (6 + 2w)(8 + 2w)
2. 48 + 72