A rectangular garden has dimensions of 10 m by 16 m and is surrounded by a wall of uniform width. If the area of the wall is [tex]120 \, \text{m}^2[/tex], find the width of the wall.

The width of the wall surrounding the rectangular garden is 2 meters. This is calculated by expressing the total area of the garden with the wall as a quadratic equation and solving for the width 'w'.

Explanation:

The question involves finding the uniform width of a wall surrounding a rectangular garden with given dimensions and area of the wall. We are given a rectangular garden with dimensions of 10m by 16m, and it is surrounded by a wall of uniform width. We are asked to find the width of the wall, given that the area of the wall is 120m2. To solve this, let us denote the width of the wall as 'w' meters.

The overall dimensions of the garden including the wall becomes (10 + 2w)m by (16 + 2w)m. The total area of the garden and the wall is given by this product:
(10 + 2w) * (16 + 2w)

The area of just the garden, without the wall, is 10 * 16 = 160m2. The additional area attributed to the wall is 120m2, which makes the total area of the garden with the wall 160m2 + 120m2 = 280m2.

Setting up the equation:
(10 + 2w) * (16 + 2w) = 280
160 + 20w + 32w + 4w2 = 280
4w2 + 52w + 160 = 280

Subtract 280 from both sides to get a quadratic equation:
4w2 + 52w - 120 = 0

Divide the entire equation by 4:
w2 + 13w - 30 = 0

Now we can factor the quadratic equation:
(w + 15)(w - 2) = 0

Setting each factor equal to zero gives us the possible values of 'w':
w + 15 = 0 or w - 2 = 0
w = -15 or w = 2

Since the width cannot be negative, we discard w = -15 and the width of the wall is 2 meters.

This type of problem is common in algebra and geometry, where solving quadratic equations and understanding geometric properties are key skills.

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