Question number three. If A, B, and C denote the lengths of the curves in the picture, labeled here A, B, and C, then which of the following inequalities is correct? In other words, we have to order the lengths of these curves from least to greatest. So let's just count. Let's count their lengths. We have for curve A, that one consists of 1, 2, 3, 4, 5, 6, 7, and then 8 segments, each of length 2. So that's 16 units. Then moving on, we have for curve B, 1, 2, 3, and 4 segments of length 2, plus what we see here are two halves of a circle, so a circle of radius 1. So we know the formula for circumference. The 4 segments of length 2 give us 8 units, plus 2 pi is the circumference of that circle. So that is roughly 8 plus 6.18, or roughly 14.18. And finally, for curve C, what we have here is 1, 2, 3, 4, again, segments of length 2, plus 2 diagonals. Two diagonals. And these diagonals here on the picture, we see that they are the exactly the hypotenuse of a 2 by 2 right triangle, and that hypotenuse would measure the square root of 8. So we have 4 times 2 plus 2 times the square root of 8, which gives us 8 plus 4 root 2. And now we have to compare 2 pi versus 4 root 2 to decide which decide which is greater. And we see that 2 pi is greater than 4 root 2. That is the same thing as saying that 2 pi is greater than 2 root 8, or saying that after dividing by 2 on both sides that pi is greater than root 8. And we can say that's the same as saying pi is greater than 2 root 2, which it certainly is. Root 2 is less than 2. It is about, it is less than 1.5, so pi is bigger than 3. And we have that, the final order here for the answers is uh curve A is the longest, curve B is next, and curve C is the shortest. So that gives us answer E over here.

curve lengths

inequality

calculation

segments

geometry