WEBVTT
00:00:00.480 --> 00:00:11.200
The radius π of a sphere is given by the formula π equals three π over four π to the power of a third, where π is the sphereβs volume.
00:00:12.080 --> 00:00:23.240
Determine the difference in radius between a sphere with volume 36π and a sphere with 2304π.
00:00:24.360 --> 00:00:39.080
In order to solve this problem, weβre gonna need to find the radius of the sphere with the volume 36π and the radius of the sphere with volume 2304π.
00:00:40.120 --> 00:00:44.800
And Iβm gonna start with the sphere that has the volume 36π.
00:00:45.520 --> 00:00:51.840
So weβre gonna use the equation π equals three π over four π to the power of third.
00:00:53.080 --> 00:00:58.280
Okay, so letβs substitute our value 36π in for π.
00:00:58.880 --> 00:01:06.960
So weβre gonna get π is equal to three multiplied by 36π over four π all to the power of third.
00:01:08.040 --> 00:01:09.640
Okay, so now letβs simplify.
00:01:10.840 --> 00:01:17.240
Well, first of all, if we divide 36 by four, we get nine.
00:01:17.840 --> 00:01:26.000
And then if we actually divide 36π by π, the πs cancel out.
00:01:26.640 --> 00:01:32.640
So weβre just left with three multiplied by nine all to the power of a third.
00:01:33.840 --> 00:01:46.720
And now, what we can actually do is we can use one of our exponent rules that says that if we have π₯ to the power of one over π, this is equal to the πth root of π₯.
00:01:47.560 --> 00:01:49.520
So therefore, weβve got three multiplied by nine.
00:01:50.080 --> 00:02:02.040
So thatβs gonna be 27 and then itβs gonna be 27 to the power of third, which is the cube root of 27, which is equal to three.
00:02:02.960 --> 00:02:07.840
So great, we found the radius of our first sphere.
00:02:08.320 --> 00:02:11.480
And the radius of our first sphere is three.
00:02:12.040 --> 00:02:18.720
Now, letβs move on to the sphere with the volume 2304π.
00:02:19.280 --> 00:02:26.600
This time weβre actually gonna substitute π equals 2304π into our formula.
00:02:27.560 --> 00:02:37.080
So weβre gonna get π is equal to three multiplied by 2304π over four π all to the power of a third.
00:02:38.320 --> 00:02:40.000
Again, weβre gonna simplify.
00:02:40.800 --> 00:02:47.160
And first of all, weβre gonna divide 2304 by four.
00:02:47.720 --> 00:02:50.920
Weβre just gonna do that using this method here.
00:02:51.880 --> 00:02:55.320
So we have four is into two donβt go.
00:02:55.840 --> 00:02:57.440
So itβs zero.
00:02:58.280 --> 00:03:00.240
And then, we carry the two.
00:03:01.320 --> 00:03:07.360
And then, we see four is into 23 go five times remainder three.
00:03:08.400 --> 00:03:15.080
Then four is into 30 goes seven remainder two.
00:03:16.160 --> 00:03:21.120
Then, four is into 24 goes six times.
00:03:22.080 --> 00:03:26.880
So we have 576.
00:03:27.680 --> 00:03:33.480
And also once again, our πs cancel out because π divided by π is just one.
00:03:34.120 --> 00:03:41.640
So weβre left with π is equal to three multiplied by 576 to the power of third.
00:03:42.720 --> 00:03:53.680
So again, we use our expanded rule that tells us that π₯ to the power of one over π equals the πth root of π₯.
00:03:54.520 --> 00:04:00.320
So we get the cube root of 1728.
00:04:01.080 --> 00:04:05.600
So therefore, we get π is equal to 12.
00:04:06.680 --> 00:04:21.960
So now, we move on to the final part of the problem that says βdetermine the difference in radius between a sphere with volume 36π and a sphere with volume 2304π.β
00:04:22.640 --> 00:04:41.480
So to determine the difference, weβre gonna need 12 because thatβs the radius of our sphere with volume 2304π minus three because thatβs the radius of our sphere with volume 36π.
00:04:42.160 --> 00:05:03.560
So therefore, we can say that given the formula π equals three π over four π to the power of third, we can say that the difference in radius between a sphere with volume 36π and a sphere with 2304π is nine.