Question

Find the rate of change of the area of a square with respect to the length $z$, the diagonal of the square. what is the rate when $z=2$?

Answer 1

Step 1: Find the area of the square.

• The length of the diagonal is $z$.
• The formula for the diagonal of the square is $z=a\sqrt{2}$.

Here, $a$ is the side length.

So, $a=\frac{z}{\sqrt{2}}$

Thus, the side of the length is $\frac{z}{\sqrt{2}}$.

And the area of the square can be calculated as:

$\begin{array}{rcl}\mathrm{Area}& =& a^2\\ & =& {\left(\frac{z}{\sqrt{2}}\right)}^2\left[a=\frac{z}{\sqrt{2}}\right]\\ & =& \frac{z^2}{2}\end{array}$

Thus, the area of the square is $\frac{z^2}{2}$.

Step 2: Find the derivative of the area.

Differentiate $A$ with respect to $z$:

$\begin{array}{rcl}\frac{dA}{dz}& =& \frac{d}{dz}\left(\frac{z^2}{2}\right)\\ & \Rightarrow & \frac{dA}{dz}=\frac{2z}{2}\\ & \Rightarrow & \frac{dA}{dz}=2\end{array}$

Therefore, the rate of change of the area of a square is $\mathbf{2}$.