# Given a formula with variables, how can I turn it around to find another solution

First of all, sorry for the crappy title, I have no idea how to ask this question.

I have a formula: $$S = \left(\frac{T\cdot D\cdot C}{1000}+V^{\frac13}\right)^3$$

I now need to turn it around to find $D$, when I have $S$. I don't even know where to start. Any pointers in the right direction, or even better, the formula to find $D$ would be much appreciated.

• This has nothing to do with calculus, please read the tag description before applying a tag. – AlexR Apr 14 '14 at 9:30

## 2 Answers

You begin with the following equation:

$$S=\left(\frac{TDC}{1000}+\sqrt[3]{V}\right)^{3}$$

You then move from the last-operation on the left-hand side inwards and perform the inverse operation on both sides, for instance, the last operation to be performed is to cube everything in the brackets, so you take the cube root of both sides to give:

$$\sqrt[3]{S}=\frac{TDC}{1000}+\sqrt[3]{V}$$

We then try and isolate the $D$ term, so our next step is to remove $\sqrt[3]{V}$ from the right-hand side by subtracting $\sqrt[3]{V}$ from both sides of the equation:

$$\sqrt[3]{S}-\sqrt[3]{V}=\frac{TDC}{1000}$$

We then multiply both sides by $1000$ to give:

$$1000(\sqrt[3]{S}-\sqrt[3]{V})=TDC$$

Dividing both sides by $TC$, we get:

$$D=\frac{1000(\sqrt[3]{S}-\sqrt[3]{V})}{TC}$$

• Thanks for the very good explanation. I'm very happy someone understood what I was looking for in spite of my complete lack of mathematical vocabulary. – peirix Apr 14 '14 at 9:46
• Get rid of "move... to the other side..." and I will vote it up. I prefer the language "get rid of $\sqrt[3]{V}$ by subtracting it from both sides". Nothing is 'moving'. – JP McCarthy Apr 14 '14 at 11:38
• @JpMcCarthy I've changed the language to be more correct. – Thomas Russell Apr 14 '14 at 11:41

Given $S = (\frac{TDC}{1000} + \sqrt[3]V)^3$ right? That's just my interpretation.

$\sqrt[3]S = \frac{TDC}{1000} + \sqrt[3]V$

$\sqrt[3]S - \sqrt[3]V = \frac{TDC}{1000}$

Next, we multiply by $\frac{1000}{TC}$

Which gives us $\displaystyle D = \frac{1000(\sqrt[3]S - \sqrt[3]V)}{TC}$

Hope that helps!