Finding rates of change using derivatives implicitly

Question

I had the following question on a an assignment last week:

A rectangles width is increasing at 9cm/s and its length is decreasing at 9cm/s. $w=10$ and $l=15$. Find the rate of change of the Area, Perimeter and Diagonals of the rectangle and whether they are increasing, decreasing or constant.

I understand that to find the area for example (fyi I am not sure what the symbol for area is so I will just use $A$;

$A=l*w \space $

$\therefore \space \Delta A = \frac{d}{dA}[l*w] $

$\frac{dl}{dA}*w+\frac{dw}{dA}*l=(9)(10)+(-9)(15)=-45\text{cm/s}^2$

I want to know how this is an implicit differentiation problem as applies to the perimeter and the diagonals because whenever I try to work it I get a rate of change for the perimeter that $\neq 0$ which is incorrect.

Answer 1

For the perimeter, let's denote it by $P$

\begin{align*} P&=2\ell+2w\\ \frac{dP}{dt}&=2\frac{d\ell}{dt}+2\frac{dw}{dt}\\ &= 2(-9\frac{cm}{s})+2\cdot 9\frac{cm}{s}\\ &=0 \end{align*}

Note also that you wrote $\frac{dl}{dA}$ and similar in your question. This is wrong, you are not taking the derivative with respect to $A$, but with respect to time $t$.