Area World problems & Factoring

Question

A rectangular garden that is 3 meters wide and 5 meters long is to has a uniform border. The area of the border is 20 square meters. What is the width of the border?

Answer 1

As Iris points out, the width is 1 m. Here's another way.

Visualize one rectangle inside another. Smaller rectangle is the garden, surrounded by the border or path.

Create 4 small squares at each corner. Their sides would all be (w) meters, so area of each is w² and there are four of them, so total area is 4w².

Now there are two remaining rectangles on the short side of the garden and two on the long side.

The path's rectangles on the short side are 3w in area, total is 3w + 3w = 6w.

The path's rectangles on the long side are 5w each, total area = 10w

We know that the total path area is 20 m². So 4w² + 6w + 10w = 20

Combining terms: 4w² + 16w = 20 or 4w² + 16w - 20 = 0 You can see that only w = 1 will work, but you can prove this by continuing to solve: Factor out the 4, then factor the trinomial -> 4(w² + 4w - 5) = 0 ->

4(w + 5)(w - 1) = 0 The only real answer for this problem is w - 1 = 0, so the path's width is 1 m.

Answer 2

The area of the concrete path needs to be measured in four parts, or 4 different rectangles. The first will be the ones that are on the long side, the side measuring 5 meters. We can say that we want it to be 6 meters long, and 1 meter wide. This gets multiplied by two, as it will be the same on the other side of the pool. These two areas together would be equal to 12 square meters. We have 8 square meters left for the rest of the pool, so we can divide that by 2. This gives us the area that we can have on each side of the remaining pool deck. Dividing 8 by 2 gives us 4 square meters. If the path is 1 meter wide, it needs to be 4 meters long to fit this.

In conclusion, the path on each long side would be 6x1, and on the other sides would be 4x1. 6+6+4+4 gives us 20.

Hope this helps!