 In this video, we provide the solution to question number 15 for the practice exam number one for Math 1060. And this one's definitely a fun one. Who doesn't love Pac-Man, right? So we see the picture of the iconic Pac-Man from the classic video game right here. And so, you know, he sees super high graphic right here. He's basically a circle with a portion removed from him. So if you see Pac-Man as this circle, then his mouth forms a central angle of said circle. And that, when his mouth is fully opened, measures to be 55 degrees. And so on our screen, you know, on the arcade game when we played Pac-Man, if his mouth has a radius of four millimeters, we then could use this information to find the exact area of Pac-Man when his mouth, of course, is all the way open. I mean, as you know, Pac-Man, he kind of opens and closes his mouth as he's eating things. So we'll take his open mouth in this situation. So how much yellow is gonna be on the screen? Clearly, I've enlarged the picture for the sake of this video here. There's two ways you could approach this problem. One is we could find the angle theta that gives all of this open area. So basically you take 306 degrees minus 55 degrees. You can go from there. I'm gonna just think of it more this way. The total area of Pac-Man, the total area of the circle would be area is equal to pi R squared, right? So this is gonna be pi times four squared. This would give us 16 pi millimeters. And we're not gonna worry, millimeters squared, excuse me. So we're not gonna worry about approximating this thing yet. We're just going to, of course, look at the area of Pac-Man for the whole circle. And then we're gonna subtract from it the area of his mouth. And so for this area formula, of course, we're gonna be using the formula we've used before, the area of a sector, the area of a pizza slice is one half theta R squared, or R squared theta, however you prefer to put it. Where R is gonna be the radius, we know that. So I'm just gonna put the radius in there of a four. So we're gonna get four squared over two theta. Four squared as we saw was 16 over two. It's gonna be eight times the angle there. And the angle does need to be in radians. So that's a very important part of this problem. So it's given to us in degrees we have to shift it over into radians. So if theta is equal to 55 degrees, we're gonna multiply that by pi over 180 degrees, for which 55 and 180 do have a common factor of five. 55 is five times 11. So you get 11 pi on top. And 180 is five times 36. So in terms of radian measure, 55 degrees is equivalent to 11 pi over 36. So put that in for theta right there. You're gonna get eight times 11 pi over 36, which again, we wanna keep things as exact as possible, right? We do have a common factor. Let's see, eight and 36, both are divisible by four. So you're gonna get two times 11 on the top, which is 22 pi. And then in the bottom, you cancel out a four. They'll leave behind a nine. So the area of the sector, so the area of his mouth is 22 pi over nine. That's not the area of Pac-Man. So we need to take the total area, which is 16 pi millimeter squared. We're gonna subtract from that the 22 pi over nine. So that's where we need to finish this thing. We're gonna take 16 pi minus the 22 pi over nine. Like so, I'm gonna take 16 times nine, which is 144, so that I have a common denominator is also it's over nine. And then 144 take away 22, that's of course gonna be 122 pi over nine. And this would be millimeter squared. That would be the exact area of Pac-Man, in which case we're gonna also approximate that. This last steps of course is not necessary for the test. The exact answer is actually the preferred answer for full credit. But if you did put that in your calculator, you would end up with 42.586 millimeter squared for the area of Pac-Man.