# Math Help - Optimizations problem with derivatives

1. ## Optimizations problem with derivatives

Question: (Attached) 59 b and c (I did a).

b: About 8.5 inches by 2 inches.
c: 20/sqrt(3) inches by 20*sqrt(2/3) inches

My work is also attached for #59 a (Page 12/14 of pdf file) which I did successfully. I cannot complete b to get to c and for b my problem is that, I cannot find a relationship between the two variables to differentiate a function with one variable. As for #59 b, I thought I was on to something and wrote some unnecessary (I think) things.

Any help would be greatly appreciated!

2. I'm not sure if I'm correct, but after several mistakes I derived an answer for the width of the plank slightly different from $8.5\,\mbox{in}$. I used the equations

\begin{aligned}
y&=\sqrt{100-x^2}-\sqrt{10}\\
A&=2xy=2x(\sqrt{100-x^2}-\sqrt{10}).
\end{aligned}

Differentiating, I obtained

\begin{aligned}
A'&=2\sqrt{100-x^2}+2x\cdot\frac{1}{2}(100-x^2)^{-\frac{1}{2}}\cdot(-2x)-2\sqrt{10}\\
Adding $2\sqrt{10}$, multiplying both sides by $\sqrt{100-x^2}$, and squaring gives us a quadratic equation from which I derived my final answer.