Using a calculating utility’s numeric solver to solve $100 = 2a \sinh (\frac{45}{a})$

Question

I've tried to use the numerical calculator (Sagemath) but can't get the right formulas.

So I tried to solve it manually using $\sinh (x)=\frac{e^x-e^{-x}}{2}$, but couldn't nut it out there.

I strongly suspect that I'm missing something obvious, so any help gratefully received.

[The answer for a is $a ≈ 56.01$ BTW]

Answer 1

If you're looking for a reference, I'd highly recommend Wolfram Alpha. You can just type in "100 = 2a*sinh(45/a)" and get an answer, both in terms of specific functions AND numerically.

https://www.wolframalpha.com/

Answer 2

Did you specifically have to use a solving machine without using your mind? If so, please don't read this answer.

First, make the equation easy enough to hold in your head.

$$100=2a\sinh\frac{45}a$$ amounts to $$\frac{50}a=\sinh\frac{45}a$$ so substitute $x=45/a$ and you get $$x=0.9\sinh x$$ which is easier on the eye and the brain.

Draw a graph of $y=x$ and $y=0.9\sinh x$ and you'll see what you're aiming at. That always helps. There is a solution at $x=0$, which corresponds to $a=\infty$, and there is another solution before $x=2$.

At this point you can use a machine if you really insist, or you can use Newton's method to solve $0.9\sinh x-x=0$. When I did this starting with $x=2$, I got the following values of $x$ on my calculator: $1.47, 1.11, 0.90, 0.817, 0.80376, 0.80344, 0.80343602804$. That is taking you to the same answer as the machine.