WEBVTT 1 00:00:00.256 --> 00:00:02.560 Welcome to Brainfuse! 2 00:00:03.840 --> 00:00:09.984 In this lesson we'll work on finding the volume of cones and spheres. If you can already 3 00:00:10.240 --> 00:00:16.384 find the volume of a right circular cylinder, you're ready for this lesson. If not, please make sure to try that video, 4 00:00:16.640 --> 00:00:22.784 do the practice problems, and come back ready to work with some more geometric solids. 5 00:00:23.296 --> 00:00:28.160 This question asks us to find the volume of a cone and sphere. 6 00:00:28.416 --> 00:00:34.048 On the screen, we have a cone, or something resembling one, and that has 7 00:00:34.560 --> 00:00:40.448 Both a radius of 6 inches and let's also add a height of 6 inches, this will be important. 8 00:00:40.704 --> 00:00:47.328 And we have a hemisphere, or half of a sphere, which has a radius of 6 inches here, 9 00:00:47.328 --> 00:00:52.992 And since all points in a true sphere are the same distance from the center it also has 10 00:00:53.248 --> 00:00:54.272 A height of 6 inches. 11 00:00:56.576 --> 00:00:57.856 Okay, 12 00:00:58.112 --> 00:01:01.184 We have the right circular cylinders below them 13 00:01:01.952 --> 00:01:07.453 With heights of 6 inches as well. 14 00:01:08.096 --> 00:01:13.472 So I'm going to pretend to fill the cone and a hemisphere with water. 15 00:01:17.056 --> 00:01:23.200 If we were to pour that water into the cylinders beneath each shape how much do you think 16 00:01:23.200 --> 00:01:26.016 Each one would hold? We can start with the cone. 17 00:01:26.784 --> 00:01:28.832 What fraction of the cylinder 18 00:01:29.088 --> 00:01:31.648 Will be filled once we pour the water 19 00:01:32.416 --> 00:01:34.208 From the cone into the cylinder? 20 00:01:40.864 --> 00:01:44.704 It would be one-third. 21 00:01:44.960 --> 00:01:48.288 Here, I'll try to make it a little more realistic. 22 00:01:48.800 --> 00:01:50.592 So it would fill 23 00:01:51.872 --> 00:01:53.152 One-third 24 00:01:53.920 --> 00:01:55.456 Of the cylinder. 25 00:01:55.712 --> 00:01:57.760 And that tells us that 26 00:01:58.016 --> 00:02:00.320 The volume of a cone 27 00:02:00.576 --> 00:02:01.856 Is equal to 28 00:02:02.112 --> 00:02:02.112 29 00:02:02.112 --> 00:02:08.768 One-third of the volume of a right circular cylinder, which is 30 00:02:08.768 --> 00:02:08.768 31 00:02:09.024 --> 00:02:13.120 Pi r squared times height. 32 00:02:13.120 --> 00:02:13.120 33 00:02:13.632 --> 00:02:16.704 So volume of a cone is equal to 34 00:02:16.960 --> 00:02:19.008 Pi r squared, 35 00:02:19.104 --> 00:02:20.800 Times the height, 36 00:02:21.312 --> 00:02:23.104 Multiplied by 1/3. 37 00:02:24.128 --> 00:02:25.152 So when we do that, 38 00:02:25.664 --> 00:02:26.688 We have 39 00:02:26.944 --> 00:02:28.480 One third 40 00:02:29.504 --> 00:02:30.528 Times 41 00:02:31.296 --> 00:02:33.600 Pi times 36 42 00:02:33.856 --> 00:02:35.136 Times 6. 43 00:02:35.392 --> 00:02:37.696 That's equal to one third 44 00:02:37.952 --> 00:02:40.512 Times 216 pi 45 00:02:45.376 --> 00:02:45.376 46 00:02:45.376 --> 00:02:50.496 Or, if you multiply that, if you make pi 3.14, 47 00:02:50.752 --> 00:02:55.104 That is one third of 678.24, 48 00:03:00.736 --> 00:03:10.208 And that equals 226.08 cubic inches. 49 00:03:10.208 --> 00:03:10.208 50 00:03:20.448 --> 00:03:23.776 Okay, so we're able to find that by using 51 00:03:24.032 --> 00:03:30.176 This formula for the volume of a cone. And now let's move to 52 00:03:30.432 --> 00:03:35.489 The sphere, or the hemisphere since it's just half of that sphere, 53 00:03:35.489 --> 00:03:40.821 And when we pour this water into the cylinder, how much of the cylinder will be filled? 54 00:03:52.547 --> 00:04:01.681 When we do that, we have two thirds of the cylinder filled. 55 00:04:14.912 --> 00:04:20.863 Not quite as accurate, but what that is supposed to show is that two-thirds of 56 00:04:21.375 --> 00:04:23.680 the cylinder will be filled. 57 00:04:25.472 --> 00:04:28.032 And that tells us that the volume 58 00:04:28.288 --> 00:04:31.616 Of a hemisphere, I'm going to put it up here on purpose, 59 00:04:32.128 --> 00:04:33.664 So "Volume-hemi," 60 00:04:33.920 --> 00:04:36.224 Equals 2/3 61 00:04:36.480 --> 00:04:40.064 Pi r squared times h. 62 00:04:40.064 --> 00:04:40.064 63 00:04:40.576 --> 00:04:45.952 And since we know that in a true sphere all points are the same distance from the center 64 00:04:46.464 --> 00:04:51.328 That means h is equal to r, h (the height) is equal to 6 inches, 65 00:04:51.584 --> 00:04:53.376 So you may also write it 66 00:04:53.888 --> 00:04:58.636 As 2/3 pi r cubed. 67 00:04:59.520 --> 00:05:03.872 But remember this is a hemisphere, and our question asks us to find 68 00:05:04.128 --> 00:05:06.176 The volume of a whole sphere. 69 00:05:06.688 --> 00:05:06.688 70 00:05:06.688 --> 00:05:09.760 So, all hope is not lost because we can still 71 00:05:10.784 --> 00:05:14.368 Use this equation to 72 00:05:14.368 --> 00:05:14.368 73 00:05:14.368 --> 00:05:14.368 74 00:05:15.136 --> 00:05:17.696 Help us figure out the volume of 75 00:05:17.952 --> 00:05:23.653 A whole sphere. Since "hemi-" is half, that means a whole sphere would have a volume 76 00:05:23.653 --> 00:05:26.144 That's double whatever this hemisphere can hold. 77 00:05:26.400 --> 00:05:28.960 So all we do is multiply 78 00:05:29.728 --> 00:05:30.752 The formula 79 00:05:31.264 --> 00:05:33.312 For the hemisphere by 2 80 00:05:39.456 --> 00:05:42.272 This is "Volume-sphere," 81 00:05:42.272 --> 00:05:42.272 82 00:05:43.808 --> 00:05:45.344 And that gives us 83 00:05:45.600 --> 00:05:51.232 4/3 pi r squared times h or 84 00:05:51.232 --> 00:05:51.232 85 00:05:52.000 --> 00:05:56.299 4/3 pi r cubed, since h is equal to r. 86 00:05:57.888 --> 00:05:59.168 They're all six inches. 87 00:05:59.424 --> 00:06:05.568 Okay, when we work on that, r cubed gives us 216. We multiply that by 88 00:06:05.824 --> 00:06:10.688 pi and that is about 678.24. 89 00:06:10.944 --> 00:06:11.968 So we have 90 00:06:12.992 --> 00:06:23.042 4/3 times 678.24 cubic inches equals 91 00:06:23.042 --> 00:06:23.042 92 00:06:23.042 --> 00:06:27.840 904.32 cubic inches. 93 00:06:31.858 --> 00:06:38.384 Okay so by comparing the volume of a cylinder to the volume of a cone 94 00:06:38.384 --> 00:06:44.081 And a hemisphere, we were able to figure out formulas for their volumes. 95 00:06:44.081 --> 00:06:51.255 You can go ahead and test these formulas by doing the practice problems. And thank you for watching this lesson.