Question Video: Changing the Subject of a Formula Mathematics

Video Transcript

The formula to calculate the volume of a sphere is 𝑉 equals four-thirds 𝜋𝑟 cubed. Make 𝑟 the subject. There’s also a second part to this question. Find the radius of a sphere with a volume of 900 cubic centimeters. Give your answer accurate to two decimal places.

Well, when we’re trying to change the subject to a formula, what we need to do is carry out inverse operations. But before we do that, what I’m going to do is rewrite our formula in a way that will make the next stage easier. And we can rewrite it like this. 𝑉 is equal to four 𝜋𝑟 cubed over three.

And if you think about why we can do that and why can we rewrite in this way. If we’re multiplying four-thirds by and then we’ve got 𝜋 over one, cause we’re gonna turn them all into fractions, then multiply it by 𝑟 cubed over one, then what this would mean is we’d multiply the numerators, say four 𝜋𝑟 cubed. And then we’d multiply the denominators, three by one by one which would give us three. So, that’s how we got our four 𝜋𝑟 cubed over three.

So, now we can carry out our inverse operations. The first of which is to multiply each side of the equation by three. And when we’ve done that, we’ll have three 𝑉 is equal to four 𝜋𝑟 cubed. So, now we’re gonna divide each side of the formula by four 𝜋. And we’re gonna do that because we want to make 𝑟 the subject, so we want 𝑟 on its own. And if we divide four 𝜋𝑟 cubed by four 𝜋, we’re gonna be left with 𝑟 cubed. And we need to remember that whatever we do to one side of the formula, we must do to the other.

So, now we’ve got three 𝑉 over four 𝜋 is equal to 𝑟 cubed. So, great, have we finished because 𝑟 is on its own? Well, no, because we want to make 𝑟 the subject of the formula, not 𝑟 cubed. So, we need to carry out one more step. And this final step is to complete the inverse operation of cubing 𝑟, and that is to take the cube root. So, we’re gonna take the cube root of both sides of our formula. And when we do that, we’re gonna get the cube root of three 𝑉 over four 𝜋 is equal to 𝑟.

So therefore, we can say that if we make 𝑟 the subject to the formula, and the formula is 𝑉 equals four thirds 𝜋𝑟 cubed, then 𝑟 is equal to the cube root of three 𝑉 over four 𝜋.

So, now for the second part of this question, we’re asked to find the radius of a sphere when we’ve been given the volume. And that volume is 900 cubic centimeters. Well, we can use the formula that we found in part one. And that’s because 𝑟 is now the subject. So, now we can also write down our other information. And that is that the volume is equal to 900. And we can substitute this into our formula to find 𝑟, the radius.

So, we put 900 into our formula. We’re gonna get 𝑟 is equal to the cube root of three multiplied by 900 over four 𝜋. And this can be rewritten as 𝑟 is equal to the cube root of 2700 over four 𝜋. That’s because three multiplied by 900 is 2700 cause three multiplied by nine is 27, then we have two zeros.

Now it’s worth reminding ourselves that we could put this into calculator as it is. Or we could use this rule to help us put it into the calculator in another way. So, we know that the cube root of 𝑎 over 𝑏 is the same as the cube root of 𝑎 divided by the cube root of 𝑏. So, you could either put the whole sort of the expression into your calculator as we have it, or you could put the cube root of 2700 divided by the cube root of four 𝜋.

So, when we work this out by calculator, we get 𝑟 is equal to 5.989418137. But we haven’t finished there. And that’s because the question asked us to give our answer accurate to two decimal places. So therefore, we count down two decimal places. And that gives us our eight. So, I’ve drawn a line after that. Then our deciding number is the nine, which is the digit to the right of our eight. And as that’s five or higher, it means that we’re gonna round our eight up. So, it’s gonna round up to a nine. So therefore, we can say that the radius of a sphere with a volume of 900 cubic centimeters is going to be 5.99 centimeters, and that’s to two decimal places.