This is the Planck length $\ell_P = \sqrt{\dfrac{\hbar G}{c^3}}$ (with $h$ in place of $\hbar$ up to a constant).

Dimensions: $[h] = ML^2T^{-1}$, $[G] = M^{-1}L^3T^{-2}$, $[c] = LT^{-1}$

$\left[\dfrac{hG}{c^3}\right] = \dfrac{ML^2T^{-1} \cdot M^{-1}L^3T^{-2}}{L^3T^{-3}} = \dfrac{L^5T^{-3}}{L^3T^{-3}} = L^2$

$\Rightarrow \sqrt{\dfrac{hG}{c^3}} \sim [L]$ ✓

Density ($\rho$) is given by: $\rho = \frac{M}{L^3}$

The relative error formula for density is:

$\frac{\Delta \rho}{\rho} \times 100 = \frac{\Delta M}{M} \times 100 + 3 \left( \frac{\Delta L}{L} \times 100 \right)$

Substitute the given values:

$\text{Max Error} = 1.5\% + 3(1\%) = 4.5\%$

John Locke's 'Two Treatises of Government' laid the groundwork for modern liberal democracy.

Compare $y = \alpha x - \beta x^2$ with $y = x \tan\theta - \frac{gx^2}{2u^2 \cos^2\theta}$.

• $\tan\theta = \alpha \implies \theta = \tan^{-1}\alpha$.
• Max height occurs at $x = \frac{\alpha}{2\beta}$ (where $dy/dx = 0$).
• $H = \alpha(\frac{\alpha}{2\beta}) - \beta(\frac{\alpha}{2\beta})^2 = \frac{\alpha^2}{2\beta} - \frac{\alpha^2}{4\beta} = \frac{\alpha^2}{4\beta}$.

This reveals that financial markets often prioritize short-term cost reduction (immediate expense savings from layoffs) over long-term human capital development, even though chronic job insecurity causes stress and can lower performance and productivity.