Moles of H$_2$O $= \dfrac{0.72}{18} = 0.04\ \text{mol}$ → H atoms $= 0.08\ \text{mol}$

Moles of CO$_2 = \dfrac{3.08}{44} = 0.07\ \text{mol}$ → C atoms $= 0.07\ \text{mol}$

Ratio C : H $= 0.07 : 0.08 = 7 : 8$

Empirical formula: $\mathbf{C_7H_8}$ (like toluene)

These two phenomena are the optical and perceptual foundations of cinema:

• Phi Phenomenon: When a series of slightly different still images is shown in rapid succession, the human brain perceives movement in the transition between images, even though no real movement exists. The brain “fills in” the gap.
• Persistence of Vision: The human eye retains a visual image for a brief fraction of a second after the source is removed. This means the interval between successive film frames appears to the brain as a continuous image rather than a series of individual snapshots.

Together, these two phenomena create the complete illusion of motion in cinema. Motion picture projectors show 24 frames per second, each flashed twice, making the transitions invisible. Eadweard Muybridge's famous 1877 photographic experiment — settling a bet about whether a galloping horse ever had all four feet off the ground — inadvertently demonstrated these principles and laid the foundation for the invention of cinema.

From the condition $|f(x)-f(y)|\le 2|x-y|^{3/2}$, divide both sides by $|x-y|$:

$\left|\frac{f(x)-f(y)}{x-y}\right|\le 2|x-y|^{1/2}$

Taking $y\to x$: $|f'(x)|\le 2\cdot0=0$, so $f'(x)=0$ for all $x$.

Hence $f$ is constant. Since $f(0)=1$, we have $f(x)=1$ for all $x$.

$\int_0^1 f^2(x)\,dx = \int_0^1 1\,dx = \mathbf{1}$