# Problem Solving Question - A sidewalk being built around a garden

We have a garden that measures $$17$$ feet by $$20$$ feet. We want to pour cement for a $$3$$-foot-wide sidewalk around the garden. To make the forms for the cement, we will need to buy some $$2$$-by-$$4$$-inch lumber. How many feet of lumber will we need just for the perimeter of the walk? (Consider both the inside and outside perimeter.)

### My solution:

I drew a picture of a garden and a sidewalk being built around the outside. The height of the inside (garden) was $$17$$ feet, and the width was $$20$$ feet. Then, I made a $$3$$-foot corner around all four corners of the garden. So, the height of the exterior was $$17 + 3 + 3 = 23$$ feet, while the width was $$3 + 3 + 20 = 26$$ feet. So, the perimeter of the inside is $$74$$ feet, and the perimeter of the outside is $$98$$ feet. I added these two to get $$172$$ feet as the total perimeter. Inexorably, I deemed that $$172$$ feet of lumber was needed for the perimeter of the walk. Is that safe to assume or am I misinterpreting the question/what it is asking for? I am getting a bit "tripped up" of the fact that the problem gave me that, "to make the forms for the cement, we will need to buy some $$2$$-by-$$4$$-inch lumber."

If anyone else cares to work out the problem/verify my solution, that would be nice.

• 2x4 lumber can actually be any length; the 2 and 4 refer to width and thickness. So, you're right. Commented Feb 13, 2012 at 23:05
• Okay, thanks. Just wanted a quick check.
– Joe
Commented Feb 13, 2012 at 23:12
• Depends: are you meant to assume that you can subtract a total of 8 inches from the inside and 8 inches from the outside due to the thickness of the planks? Commented Feb 13, 2012 at 23:14
• Hard to know, depends on how you handle corners. Might need $2$ extra inches at each corner, actually not quite $2$ since a two by four is less than $2$ inches thick. Commented Feb 13, 2012 at 23:14
• In all likelihood yours is the desired answer, if this is a textbook question. As a couple of the commenters have noted, a practical real-world answer would have to take a bit more into account. Commented Feb 13, 2012 at 23:24

So the sum of the interior and exterior perimeters of the concrete is $2 \times 17 + 2 \times 20 + 2 \times 23 + 2 \times 26 = 172 \text{ feet}.$
So long as the edging has constant width $w$ (2 inches, 4 inches, something else), this remains true; what you save on the inside, you need on the outside. So the length needed as shown in the diagram is $2 \times (17-2w) + 2 \times 20 + 2 \times 23 + 2 \times (26+2w) = 172 \text{ feet}$ again.