To determine if the height of the mountain can be calculated, we use the relationship between the time lost by a seconds pendulum and the height above the Earth's surface. A seconds pendulum has a time period of $T = 2 \text{ s}$ at the base of the mountain. Given: Time lost per hour, $\Delta t = 30 \text{ s}$ Total time in one hour, $t = 3600 \text{ s}$ Radius of the Earth, $R = 6400 \text{ km} = 6.4 \times 10^6 \text{ m}$ The time period of a pendulum at the surface is $T = 2\pi \sqrt{\frac{L}{g}}$ and at height $h$ is $T' = 2\pi \sqrt{\frac{L}{g'}}$. The relationship between the time lost $\Delta t$ and the height $h$ is given by: $$\frac{T'}{T} = \frac{t}{t - \Delta t}$$ Since $g' = g \left(1 + \frac{h}{R}\right)^{-2} \approx g \left(1 - \frac{2h}{R}\right)$, we have $\frac{T'}{T} = \left(1 - \frac{2h}{R}\right)^{-1/2} \approx 1 + \frac{h}{R}$. Equating the expressions: $$ \begin{aligned} 1 + \frac{h}{R} &= \frac{t}{t - \Delta t} \\ \frac{h}{R} &= \frac{t}{t - \Delta t} - 1 \\ \frac{h}{R} &= \frac{t - (t - \Delta t)}{t - \Delta t} \\ \frac{h}{R} &= \frac{\Delta t}{t - \Delta t} \end{aligned} $$ Substituting the values: $$ \begin{aligned} h &= R \times \frac{\Delta t}{t - \Delta t} \\ h &= 6400 \times \frac{30}{3600 - 30} \\ h &= 6400 \times \frac{30}{3570} \\ h &= 6400 \times 0.008403 \\ h &\approx 53.78 \text{ km} \end{aligned} $$ Since all necessary variables are provided, it is mathematically possible to determine the height of the mountain. The calculated height is approximately $53.78 \text{ km}$.