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.
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.