1 00:00:10,677 --> 00:00:13,661 ...and they tell us that p is greater than 7r. 2 00:00:13,661 --> 00:00:17,128 So, let's first think about the area of a rectangle 3 00:00:17,128 --> 00:00:19,698 with length p and width 2r. 4 00:00:19,698 --> 00:00:23,079 So, this is our rectangle right here... 5 00:00:23,079 --> 00:00:28,390 it has a length of p and it has a width of 2r. 6 00:00:28,390 --> 00:00:30,423 So, what's its area? Well it's just going to be 7 00:00:30,423 --> 00:00:34,300 the length times the width. So, the area here 8 00:00:34,300 --> 00:00:40,520 is going to be p... or maybe I should say 2rp. 9 00:00:40,520 --> 00:00:43,356 This is the length times the width, or the width times the length. 10 00:00:43,356 --> 00:00:45,669 So, the area is equal to 2rp for the rectangle. 11 00:00:45,669 --> 00:00:48,213 Now. We also want to find the difference between 12 00:00:48,213 --> 00:00:50,433 this area and the area of a circle. 13 00:00:50,433 --> 00:00:53,064 The area of a circle with diameter 4r. 14 00:00:53,064 --> 00:00:55,372 So, what's the area of the circle going to be? 15 00:00:55,372 --> 00:00:57,770 So, let me draw our circle over here... 16 00:00:57,786 --> 00:01:04,845 So, our circle looks like that. It's diameter is 4r. 17 00:01:04,845 --> 00:01:06,945 How do we figure out the area of a circle? 18 00:01:06,945 --> 00:01:14,837 The area is equal to pi(r) squared for a circle, where r is the radius. 19 00:01:14,837 --> 00:01:17,213 They gave us the diameter. The radius is half of that. 20 00:01:17,213 --> 00:01:21,264 So, the radius here is going to be half this distance or 2r. 21 00:01:21,264 --> 00:01:29,219 So, the area of our circle is going to be pi times 2r squared. 22 00:01:29,219 --> 00:01:31,463 This is the radius right there, so we're squaring the entire radius. 23 00:01:31,463 --> 00:01:37,144 So, this is going to be equal to pi times 4 times r2, I'm just squaring each of these terms. 24 00:01:37,144 --> 00:01:47,670 Or, if we were to change the order, the area of the circle is equal to 4(pi)r2. 25 00:01:47,670 --> 00:01:49,631 And we want to find the difference. 26 00:01:49,631 --> 00:01:53,630 So, to find the difference, it's helpful, just so we don't end up with a negative number... 27 00:01:53,630 --> 00:01:55,813 to figure out which of these two is larger. 28 00:01:55,813 --> 00:02:01,354 So, they're telling us that p is greater than 7r. 29 00:02:01,354 --> 00:02:05,679 So, let's think about this. If p is greater than 7r... 30 00:02:05,679 --> 00:02:08,448 then 2... let me write it this way... 31 00:02:08,448 --> 00:02:13,739 We know that p is greater than 7r, so if we were to multiply 32 00:02:13,739 --> 00:02:17,282 both sides of this equation by 2r, and 2r is positive... 33 00:02:17,282 --> 00:02:20,193 we're dealing with positive distances - positive lengths 34 00:02:20,193 --> 00:02:24,784 So, if we multiply both sides of this equation by 2r, it shouldn't change the equation. 35 00:02:24,784 --> 00:02:29,521 So, multiply that by 2r, and then multiply this by 2r. 36 00:02:29,521 --> 00:02:41,053 Then our equation becomes 2r(p) is greater than 14 r squared. 37 00:02:41,053 --> 00:02:44,402 Now why is this interesting? Actually, why did I multiply this by 2r? 38 00:02:44,402 --> 00:02:47,677 Well that's so that this becomes the same as the area of the rectangle. 39 00:02:47,677 --> 00:02:55,867 So, this is the area of the rectangle, and what's 14 r squared? 40 00:02:55,867 --> 00:03:02,306 Well, 4 times pi is going to get us something less than 14. 41 00:03:02,306 --> 00:03:09,335 This is less than 14, so this is 4 pi is less than 14. 42 00:03:09,335 --> 00:03:23,829 14 is ... let me put it this way... 4 times 3.5 is equal to 14, right? 43 00:03:23,829 --> 00:03:30,140 So, 4 times pi, which is less than 3.5, is going to be less than 14. 44 00:03:30,140 --> 00:03:35,125 So, we know that this over here is larger than this quantity over here... 45 00:03:35,125 --> 00:03:39,398 it's larger than 4(pi)r squared. And so we know that 46 00:03:39,398 --> 00:03:43,762 this rectangle has a larger area than the circle. 47 00:03:43,762 --> 00:03:46,440 So, we can just subtract the circle's area from 48 00:03:46,440 --> 00:03:48,310 the rectangle's area to find the difference. 49 00:03:48,310 --> 00:03:50,898 So, the difference is going to be the area of the rectangle 50 00:03:50,898 --> 00:03:55,099 which we already figured out as 2r(p) 51 00:03:55,099 --> 00:03:58,545 ...and we're going to subtract from that the area of the circle. 52 00:03:58,545 --> 00:04:05,976 The area of the circle is 4(pi)r squared. 53 00:04:05,976 --> 00:04:08,839 So, hopefully that made sense... 54 00:04:08,839 --> 00:04:12,516 One point I want to clarify, I gave the equation 55 00:04:12,516 --> 00:04:16,417 of a circle - the area of a circle - to be (pi)r squared. 56 00:04:16,417 --> 00:04:18,578 And then we said that the radius is actually 57 00:04:18,578 --> 00:04:22,469 2r in this case, so, I substituted 2r for r. 58 00:04:22,469 --> 00:04:23,523 Hopefully, that doesn't confuse you. 59 00:04:23,523 --> 00:04:26,453 This r is the general term for any radius. 60 00:04:26,453 --> 00:04:30,699 They later told us that the actual radius is 2 times some letter r. 61 00:04:30,699 --> 00:04:33,267 So, I substitute that into the formula. 62 00:04:33,267 --> 99:59:59,999 Anyway, hopefully you found that useful.