A rectangular garden has a length that is twice its width. A uniform pathway of width 3 m running around the outside of the garden has an area of 288 m². What is the length of the garden (in meters)?

The correct answer is 28.

Let us break down the solution step-by-step:

Step 1: Define the dimensions of the garden.

Let the width of the rectangular garden be $w$ meters.

Since the length of the garden is twice its width, the length of the garden $l$ is:

$l=2w$

The area of the rectangular garden is:

$\text{Garden Area}=\text{length}\times \text{width}=2w\times w=2w^2$

Step 2: Determine the outer dimensions including the pathway.

A uniform pathway of width 3 m surrounds the garden on all sides. Therefore, 3 m is added to both ends of the length and width:

$\text{Outer Length}=2w+3+3=2w+6$

$\text{Outer Width}=w+3+3=w+6$

Step 3: Express the area of the pathway.

The total area of the outer region (garden plus pathway) is:

$\text{Outer Area}=(2w+6)(w+6)$

Expanding the expression:

$\text{Outer Area}=2w^2+12w+6w+36=2w^2+18w+36$

The area of the pathway alone is the difference between the total outer area and the garden area:

$\text{Pathway Area}=\text{Outer Area}-\text{Garden Area}$

$\text{Pathway Area}=(2w^2+18w+36)-2w^2=18w+36$

Step 4: Solve for width ($w$).

We are given that the pathway area is 288 m²:

$18w+36=288$

Subtract 36 from both sides:

$18w=288-36$

$18w=252$

Divide by 18:

$w=\frac{252}{18}=14$

So, the width of the garden is 14 m.

Step 5: Calculate the length of the garden.

Since the length is twice the width:

$\text{Length}=2\times 14=28$

Thus, the length of the garden is 28 meters.