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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.

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  • $\begingroup$ 2x4 lumber can actually be any length; the 2 and 4 refer to width and thickness. So, you're right. $\endgroup$ Commented Feb 13, 2012 at 23:05
  • $\begingroup$ Okay, thanks. Just wanted a quick check. $\endgroup$
    – Joe
    Commented Feb 13, 2012 at 23:12
  • $\begingroup$ 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? $\endgroup$ Commented Feb 13, 2012 at 23:14
  • $\begingroup$ 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. $\endgroup$ Commented Feb 13, 2012 at 23:14
  • $\begingroup$ 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. $\endgroup$ Commented Feb 13, 2012 at 23:24

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Your answer is correct: consider the following diagram.

enter image description here

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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  • $\begingroup$ Nice diagram. Makes sense - slightly quicker/easier way to the solution! $\endgroup$
    – Joe
    Commented Feb 13, 2012 at 23:54

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