# related rates what units to use when rate given is in different units to final answer

The length of a particular rectangle is four times its width and this ratio is maintained as the width is increased at 2 mm/sec. Find the rate of increase in the area of the rectangle when the width is 15 cm.

So I get the rate of increase of area to be 16w (w=width), keeping the rate of width increase in millimetres. I than convert 15cm also into millimetres to keep things consistent, but when I solve it:

16*150=2400mm/sec

I get the wrong answer (240mm/sec). If however, I do everything in cm, I get the correct answer and I can't figure out why. I think it has to do with the differentiation of the area with respect to width having associated units, but if that's the case, I'm really confused as to why.

Since $l = 4w$, the area $A = lw$ can be expressed as a function of $w$. $$A(w) = (4w)w = 4w^2$$ Differentiating with respect to time yields $$\frac{dA}{dt} = 8w \frac{dw}{dt}$$ If we work with $\text{cm}$, we obtain $$\frac{dA}{dt} = 8(15~\text{cm})\left(0.2~\frac{\text{cm}}{\text{s}}\right) = 24~\frac{\text{cm}^2}{\text{s}}$$ If we work with $\text{mm}$, we obtain $$\frac{dA}{dt} = 8(150~\text{mm})\left(2~\frac{\text{mm}}{\text{s}}\right) = 2400~\frac{\text{mm}^2}{\text{s}}$$ Since there are $10~\text{mm}$ in $1~\text{cm}$, there are $(10~\text{mm})^2 = 100~\text{mm}^2$ in $1~\text{cm}^2$, so the answers are equivalent.