WEBVTT
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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.
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Determine the difference in radius between a sphere with volume 36π and a sphere with 2304π.
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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.
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And Iβm gonna start with the sphere that has the volume 36π.
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So weβre gonna use the equation π equals three π over four π to the power of third.
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Okay, so letβs substitute our value 36π in for π.
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So weβre gonna get π is equal to three multiplied by 36π over four π all to the power of third.
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Okay, so now letβs simplify.
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Well, first of all, if we divide 36 by four, we get nine.
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And then if we actually divide 36π by π, the πs cancel out.
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So weβre just left with three multiplied by nine all to the power of a third.
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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₯.
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So therefore, weβve got three multiplied by nine.
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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.
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So great, we found the radius of our first sphere.
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And the radius of our first sphere is three.
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Now, letβs move on to the sphere with the volume 2304π.
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This time weβre actually gonna substitute π equals 2304π into our formula.
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So weβre gonna get π is equal to three multiplied by 2304π over four π all to the power of a third.
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Again, weβre gonna simplify.
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And first of all, weβre gonna divide 2304 by four.
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Weβre just gonna do that using this method here.
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So we have four is into two donβt go.
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So itβs zero.
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And then, we carry the two.
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And then, we see four is into 23 go five times remainder three.
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Then four is into 30 goes seven remainder two.
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Then, four is into 24 goes six times.
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So we have 576.
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And also once again, our πs cancel out because π divided by π is just one.
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So weβre left with π is equal to three multiplied by 576 to the power of third.
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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₯.
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So we get the cube root of 1728.
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So therefore, we get π is equal to 12.
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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
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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.
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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.