Video: Evaluating Unknowns in Given Formulas by Direct Substitution Using Cube Roots

The radius ๐‘Ÿ of a sphere is given by the formula ๐‘Ÿ = (3๐‘‰/4๐œ‹)ยนแŸยณ, where ๐‘‰ is the sphereโ€™s volume. Determine the difference in radius between a sphere with volume 36๐œ‹ and a sphere with 2304๐œ‹.

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Video Transcript

The radius ๐‘Ÿ of a sphere is given by the formula ๐‘Ÿ equals three ๐‘‰ over four ๐œ‹ to the power of a third, where ๐‘‰ is the sphereโ€™s volume. Determine the difference in radius between a sphere with volume 36๐œ‹ and a sphere with 2304๐œ‹.

In order to solve this problem, weโ€™re gonna need to find the radius of the sphere with the volume 36๐œ‹ and the radius of the sphere with volume 2304๐œ‹. And Iโ€™m gonna start with the sphere that has the volume 36๐œ‹. So weโ€™re gonna use the equation ๐‘Ÿ equals three ๐‘‰ over four ๐œ‹ to the power of third.

Okay, so letโ€™s substitute our value 36๐œ‹ in for ๐‘‰. So weโ€™re gonna get ๐‘Ÿ is equal to three multiplied by 36๐œ‹ over four ๐œ‹ all to the power of third. Okay, so now letโ€™s simplify. Well, first of all, if we divide 36 by four, we get nine. And then if we actually divide 36๐œ‹ by ๐œ‹, the ๐œ‹s cancel out. So weโ€™re just left with three multiplied by nine all to the power of a third.

And now, what we can actually do is we can use one of our exponent rules that says that if we have ๐‘ฅ to the power of one over ๐‘Ž, this is equal to the ๐‘Žth root of ๐‘ฅ. So therefore, weโ€™ve got three multiplied by nine. So thatโ€™s gonna be 27 and then itโ€™s gonna be 27 to the power of third, which is the cube root of 27, which is equal to three. So great, we found the radius of our first sphere. And the radius of our first sphere is three.

Now, letโ€™s move on to the sphere with the volume 2304๐œ‹. This time weโ€™re actually gonna substitute ๐‘‰ equals 2304๐œ‹ into our formula. So weโ€™re gonna get ๐‘Ÿ is equal to three multiplied by 2304๐œ‹ over four ๐œ‹ all to the power of a third.

Again, weโ€™re gonna simplify. And first of all, weโ€™re gonna divide 2304 by four. Weโ€™re just gonna do that using this method here. So we have four is into two donโ€™t go. So itโ€™s zero. And then, we carry the two. And then, we see four is into 23 go five times remainder three. Then four is into 30 goes seven remainder two. Then, four is into 24 goes six times. So we have 576. And also once again, our ๐œ‹s cancel out because ๐œ‹ divided by ๐œ‹ is just one. So weโ€™re left with ๐‘Ÿ is equal to three multiplied by 576 to the power of third.

So again, we use our expanded rule that tells us that ๐‘ฅ to the power of one over ๐‘Ž equals the ๐‘Žth root of ๐‘ฅ. So we get the cube root of 1728. So therefore, we get ๐‘Ÿ is equal to 12.

So now, we move on to the final part of the problem that says โ€œdetermine the difference in radius between a sphere with volume 36๐œ‹ and a sphere with volume 2304๐œ‹.โ€ So to determine the difference, weโ€™re gonna need 12 because thatโ€™s the radius of our sphere with volume 2304๐œ‹ minus three because thatโ€™s the radius of our sphere with volume 36๐œ‹.

So therefore, we can say that given the formula ๐‘Ÿ equals three ๐‘‰ over four ๐œ‹ to the power of third, we can say that the difference in radius between a sphere with volume 36๐œ‹ and a sphere with 2304๐œ‹ is nine.

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