High School

A rectangular garden measures 10 m by 16 m and is to be surrounded by a concrete walk of uniform width. If the area of the walk is 120 m\(^2\), find the width of the walk.

Answer :

Final answer:

The question involves solving a quadratic equation to find the width of a concrete walk around a garden. The garden measures 10 m by 16 m, and the area of the walk is 120 m². By setting up an equation for the total area and factoring, the width of the walk is found to be 2 meters.

Explanation:

The question involves finding the uniform width of a concrete walk surrounding a rectangular garden. The garden measures 10 m by 16 m with the area of the walk being 120 m2. Let's call the width of the walk 'w'. The total area covered by the garden and the walk will be (10 + 2w) * (16 + 2w). Since we know the area of just the garden is 10 * 16 = 160 m2, we can find the total area by adding 120 m2 to it, resulting in 280 m2. Now we can set up the equation:

(10 + 2w) * (16 + 2w) = 280

To solve for 'w', expand the left side of the equation:

160 + 20w + 32w + 4w2 = 280

Combining like terms gives us:

4w2 + 52w + 160 = 280

Subtract 280 from both sides to set the equation to zero:

4w2 + 52w - 120 = 0

Divide everything by 4 to simplify:

w2 + 13w - 30 = 0

Factor the quadratic equation:

(w + 15)(w - 2) = 0

Therefore, we have two possible solutions for 'w': -15 and 2. Since a negative width is not possible for a walk, we discard -15 and are left with the width of the walk as 2 meters.