Diving Bell

We will start with the law of Boyle-Mariotte which was derived to $$p_1V_1 = p_2 V_2$$ Here $p_1$ is the atmospheric pressure and $V_1$ is the volume of the diving bell. The values on the left side of this equation denote the volume and pressure of the gas after the diving bell has been positioned inside the water. Since the area of the diving bell is constant, it cancels out on both sides and only the following equation remains: $$p_2 = \frac{h}{d-h}p_1$$ Now we can replace $p_2$ with the equation for the hydrostatic pressure: $$p_2 = p_1 + \varrho g (d-h)$$ This then leads to the following equation: $$p_1 + \varrho g (d-h) = \frac{h}{d-h}p_1$$ We can either solve this tricky equation manually or with the help of a computer algebra system. In both cases, we receive the following formula for $h$: $$h=-\frac{\sqrt{4 d g p1} \varrho+{p_1}^{2}-2 d g \varrho-p_1}{2 g \varrho}$$ Inserting all values results in $h = 0.94\,\mathrm{cm}$ which is close 1m.