# 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 ft 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.

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2x4 lumber can actually be any length; the 2 and 4 refer to width and thickness. So, you're right. –  El'endia Starman Feb 13 '12 at 23:05
Okay, thanks. Just wanted a quick check. –  Joe Feb 13 '12 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? –  Peter Taylor Feb 13 '12 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. –  André Nicolas Feb 13 '12 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. –  Brian M. Scott Feb 13 '12 at 23:24

## 1 Answer

Your answer is correct: consider the following diagram.

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.

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Nice diagram. Makes sense - slightly quicker/easier way to the solution! –  Joe Feb 13 '12 at 23:54