You can get right answer with your S = (6561 w^3 + 917000)/(2025 w) but my S = 36680/(81 w) + (81 w^2)/25 is a bit easier too look at.
Either way you now need to minimize S. That will involve the derivative. If you do not know what that is then we can minimize using iteration but that might be confusing also. There might be another way using the AM-GM inequality but that too would be confusing.
Looking at the book you are using that fellow uses a graphic calculator on his earlier problems. You are trying to do the problems without one of those? For that you will need a bit more math than you currently know. The graphic calculator way does not require you to know so much math. The pencil and paper way requires that you do.
A) S = 2*h*w + 2 l*h + 2 w*l
Using the fact that h = (81 / 50) w we substitute and get
B) S = (131 l w)/25 + (81 w^2)/25
Solving for l in l*w*h = 140 we get
l = 7000/(81 w^2)
Substituting that into B) we get:
S = 36680/(81 w) + (81 w^2)/25
Now all you have to do is set the derivative of that equal to 0 and solve for w. What do you get now? The problem with this approach is that you will have to solve a cubic. Easy for a computer, hard for a human. And, 4.12 is correct for w rounded to 2 decimal places.
You need to minimize the surface area 2(hw)+2(lh)+2(wl) subject to the constraints that lwh=140 and h=1.62w. This leads to a definite answer.
Oh and by the way, people see the legendary Sasquatch here often. It goes by the name Skunk Ape. No one really knows what the creature eats. Speculation says, it will grab a gator and snap it into two pieces as effortlessly as we could a twig. Others believe it kills bears for its grub. The darker side people think he will eat a human when it can. Nobody, but nobody, thinks it eats cereal.
In regards to your previous question.
There are at least 9 different definitions of empirical quantiles.
So both numpy and Mathematica etc are correct, depending on what definition the textbook is using.
Is 100 percentile possible? I think theoretically no
I would think that 100 percentile would mean the value that 100 percent of the data would be less than. But some definitions obviously include equal to also. I have seen many cases of 100 percentile computed and urge you to look at this answer:
whuber, is an expert on statistics and he seems to indicate 100 percentile is allowed.