WEBVTT 00:00:02.940 --> 00:00:12.220 Which of the following is the equation of the sphere of center eight, negative 15, 10 and passing through the point negative 14, 13, negative 14? 00:00:12.550 --> 00:00:21.320 Is it option (A) π‘₯ minus eight all squared plus 𝑦 plus 15 all squared plus 𝑧 minus 10 all squared is equal to 1,844? 00:00:21.640 --> 00:00:30.590 Is it option (B) π‘₯ minus eight all squared minus 𝑦 plus 15 all squared minus 𝑧 minus 10 all squared is equal to 1,844? 00:00:30.850 --> 00:00:38.340 Or is it option (C) π‘₯ minus eight all squared plus 𝑦 plus 15 all squared plus 𝑧 minus 10 all squared is equal to 56? 00:00:38.640 --> 00:00:47.500 Or finally, is it option (D) π‘₯ minus eight all squared minus 𝑦 plus 15 all squared minus 𝑧 minus 10 all squared is equal to 56? 00:00:47.930 --> 00:00:53.170 In this question, we’re asked to determine which of four equations represents the equation of a sphere. 00:00:53.540 --> 00:00:58.160 We’re told that the center of the sphere is the point eight, negative 15, 10. 00:00:58.200 --> 00:01:04.120 And we’re also told that this sphere passes through the point negative 14, 13, negative 14. 00:01:04.400 --> 00:01:07.260 There’re several different methods we can use to answer this question. 00:01:07.310 --> 00:01:12.210 For example, we could look at the options given to us and try to eliminate some of these options. 00:01:12.420 --> 00:01:18.210 However, by looking at these options, we can see they’re very similar to the standard form of the equation of a sphere. 00:01:18.400 --> 00:01:29.090 So we’ll answer this question by just finding the standard form of the equation of a sphere with center eight, negative 15, 10 passing through the point negative 14, 13, negative 14. 00:01:29.490 --> 00:01:42.320 We recall a sphere centered at the point π‘Ž, 𝑏, 𝑐 with a radius of π‘Ÿ, which must be positive, has the equation π‘₯ minus π‘Ž all squared plus 𝑦 minus 𝑏 all squared plus 𝑧 minus 𝑐 all squared is equal to π‘Ÿ squared. 00:01:42.610 --> 00:01:45.470 This is called the standard form of the equation of the sphere. 00:01:45.770 --> 00:01:48.390 All spheres can be represented in standard form. 00:01:48.490 --> 00:01:54.130 All we need to know to find the standard form of the equation of a sphere is its center and its radius. 00:01:54.390 --> 00:02:00.060 In this question, we’re already told that the center of the sphere is the point eight, negative 15, 10. 00:02:00.440 --> 00:02:09.440 Therefore, in the standard form of the equation of the sphere, the value of π‘Ž will be eight, the value of 𝑏 will be negative 15, and the value of 𝑐 will be 10. 00:02:09.750 --> 00:02:13.630 We can substitute these values into the standard form of the equation of the sphere. 00:02:14.040 --> 00:02:27.130 This means that the standard form of the equation of the sphere given to us in the question must be in the form π‘₯ minus eight all squared plus 𝑦 plus 15 all squared plus 𝑧 minus 10 all squared is equal to π‘Ÿ squared, where π‘Ÿ will be the radius of the sphere. 00:02:27.490 --> 00:02:33.180 All we need to find now to find the standard form of the equation of the sphere is to find the value of the radius π‘Ÿ. 00:02:33.440 --> 00:02:37.710 To find this radius, we need to recall exactly what we mean by the radius of a sphere. 00:02:37.970 --> 00:02:44.930 Remember, a sphere is a shape in three dimensions which represents all points equidistant from a point called the center. 00:02:45.330 --> 00:02:47.780 This distance is called the radius of the sphere. 00:02:48.030 --> 00:03:00.360 And since we know that our sphere passes through the point negative 14, 13, negative 14 and we know the coordinates of the center of the sphere, the distance between these two points must be equal to the radius of the sphere. 00:03:00.720 --> 00:03:05.610 Therefore, we can find the radius of this sphere by finding the distance between these two points. 00:03:05.830 --> 00:03:11.050 And to do this, we recall the formula for finding the distance between two points in three-dimensional space. 00:03:11.250 --> 00:03:27.790 The distance between the point π‘₯ sub one, 𝑦 sub one, 𝑧 sub one and the point π‘₯ sub two, 𝑦 sub two, 𝑧 sub two is given by the square root of π‘₯ sub one minus π‘₯ sub two all squared plus 𝑦 sub one minus 𝑦 sub two all squared plus 𝑧 sub one minus 𝑧 sub two all squared. 00:03:28.120 --> 00:03:32.970 And it doesn’t matter in which order we label the two points; the distance between them will be the same. 00:03:33.260 --> 00:03:51.090 Substituting the point on our sphere and the center of our sphere into the formula for the distance between two points in three-dimensional space gives us that the radius of the sphere is equal to the square root of negative 14 minus eight all squared plus 13 minus negative 15 all squared plus negative 14 minus 10 all squared. 00:03:51.460 --> 00:03:58.750 Simplifying the expression inside of our square root symbol, we get that π‘Ÿ is equal to the square root of 1,844. 00:03:59.000 --> 00:04:06.390 Now we could simplify this expression further; however, remember, we’re trying to find an expression for π‘Ÿ squared for the standard form of the equation of our sphere. 00:04:06.730 --> 00:04:10.070 This means we’re going to want to square both sides of this equation anyway. 00:04:10.460 --> 00:04:15.140 Doing this gives us that π‘Ÿ squared will be equal to 1,844. 00:04:15.360 --> 00:04:20.930 Finally, we can substitute this value of π‘Ÿ squared into the standard form of the equation of our sphere. 00:04:21.300 --> 00:04:40.770 This gives us that the standard form of the equation of a sphere centered at the point eight, negative 15, 10 passing through the point negative 14, 13, negative 14 π‘₯ minus eight all squared plus 𝑦 plus 15 all squared plus 𝑧 minus 10 all squared is equal to 1,844, which we can see is given by answer (A).