In a cylinder lying on its side one fourth full , how high is water in it?

Question

Seems like a simple question to get the formula for the area of a circle segment and to solve for h. The problem is, I have to solve this equation for x first, in :
$$ \frac{\pi}{4}=\left(\frac{r^2}{2}\right)\left(\frac{\pi}{360}x-\sin(x)\right) $$

is the fraction of area of the remaining water, the length of the cylinder is irrelevant)

but unfortunately, it seems no calculator in the world is able to solve this for x, because it has an x both inside and outside of a trigonometric function. Is there another way to solve this or, is my calculator just not good enough?

Answer 1

In radians and with $r=1$, the equation is

$$x-\sin x=\frac{\pi}4$$ or $$x=\sin x+\frac\pi4.$$

As you pointed, it has no closed-form expression as the unknown appears both inside and outside the trigonometric function.

You can obtain a numerical estimate by the fixed-point method, using a simple scientific calculator. Indeed, the iterations

$$x_{n+1}=\sin x_n+\frac\pi4$$ will converge to the root ($1.76634028\cdots$) from any starting value.

---

For the sake of illustration, here are the first four iterates of the function, clearly showing that it tends to a constant.

---

Faster convergence is obtained by Newton's iterations,

$$x_{n+1}=x_n-\frac{x_n-\sin x_n-\frac\pi4}{1-\cos x_n}.$$

Answer 2

EDIT1&2

It is a transcendental equation, many hand held calculators do the job.You have not applied radian measure throughout. Secondly after stating that $r$ does not matter for angle you brought it in again into equation.

If $t$ is half apex angle in radian measure,

FindRoot[Pi/4. == ( 2 t - Sin[2 t]) , {t, .5}]

gives $ t \approx 0.883.$