ACT 2.8 - Finding Arc Lengths

Añadido: 2026-05-13

[00:00:00]Hello and welcome. So we are at the infamous  zipper problem. Okay, I mean it's not infamous, it's just it's so entertaining. All right, so we  have a plastic cover is made for a cylindrical pool that has a diameter of 22 feet and a height  of 10 feet. All right, let's let's break this down a little bit. First off, diameter uh how does  the diameter relate to anything else in a circle right? A diameter is going to be twice the  radius, so if you want to find the radius of a circle all we take is the diameter and we  half it. So that means if our diameter is 22, half of that is going to be 11. So that  means while we're told the diameter is 22, we immediately know that our radius is 11. Okay, and  then we're told that we have a height of 10 feet, so that means that we have a cylinder right, and  this is a 10 foot height. Might be useful at the moment, we don't know. Okay, and then from  there we have a cover on top of this pool, and it will include a wedge shaped flap that  forms a 29 degree angle at the center of the cover as shown in the figure below. A zipper  will go along one side of the wedge-shaped flap and around the arc. Which of the following is  closest to the length in feet of the zipper? Okay, so first off this whole cylindrical pool  thing with 10 feet, that's useless information okay. You're trying to find the length of this  zipper. Now, does the zipper have anything to do with the height of the pool? No, we're  only talking about the surface of the pool, we don't care about how tall it is. That 10 is  a distractor number, don't let it distract you.

[00:01:38]Okay, great news is that we already know part of  the zipper. By finding our radius, we already know that that part of our zipper, this little  section right there, we know that is now 11. Huzzah. So the only thing we really have to  worry about at this point is what that length is.

[00:01:55]We're also told a couple of things. The first  thing that we're told is that we're told bam, 29 degrees is the angle forming this arc, and  you say okay well how do we find the arc itself.

[00:02:08]Well, let's use a little bit of our circle  common sense to help us out with this thing.

[00:02:12]So first thing I would say is okay um  that yellow portion. The arc is on the what we call circumference of the circle right.  Circumference is like perimeter but for a circle, it's the outside edge of a circle. Now an  arc is a portion of the circumference okay, it's just a little little section there.  Specifically, it's a 29 degree section of your circumference. So how we can find it is we could  say right well circumference of a circle is 2 pi r okay, and 2 pi r where we know our r is 11,  we already found that, that's 2 pi times 11, in other words 22 pi. That means that that entire  yellow line is 22 pi. Well we know that of that 22 pi, we're only taking 29 degrees of it okay. And  what's the all the way around the circle? What's the whole circle degree measure? 360 right. To  go all the way around a circle, it's 360 degrees.

[00:03:18]That means that if you take your circumference and  you multiply it by the amount of circle you have, that's it. This is a percentage right, you're just  taking a certain percentage of the circumference, that's all we're doing here, and that's how we can  find our arc length. So then all we do is this is just a little bit of calculator work okay. We'll  notice that none of your values here have pi, so that means we're going to be multiplying pi in  on this one, and this is just plug and chug into some calculators. So let's go into this. So we  have 22 pi times our what was it 29 degrees or 30, 29 degrees there we go, times 29 over 360. Right,  you don't have to worry about the degree symbol because when you have a degree over degree right,  these will essentially end up simplifying with each other, they'll cancel out and leave you in  just straight radians. Okay, so then you have 22 pi times 29 over 360, and we're left with  5.567600314, all right.

[00:04:31]Oh golly I forgot it, 5.568. Okay, so  approximately 5.568, there we go. Alrighty now are we done? No right, we've found this section  now to be a 5.568. Well now we're finding the length of the whole zipper. It's not just the  5.568, it's that plus the radius right, which means that we do 11 plus that number. So 11 plus  5.568, and we end up with 16.568, which is closest to one of our answers known as 17. And that is it,  that is your correct answer for this particular set of questions-- this particular question, set of  numbers in the question. All right, and that's it, that's as crazy as it gets. It really just boils  down to your circumference times the percent of circle that you have okay. That that is all  it takes to solve this problem for the length of arc that you need okay, plus whatever  the radius because this thing is the arc length. Okay, not too crazy, thanks for watching.