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(D) The pressure variation in a fluid with varying density is given by the hydrostatic law: $dP = -\rho(z) g dz$.Integrating from the bottom ($z=0$, $

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$\frac{5}{3} \rho_0 g H$

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(D) The pressure variation in a fluid with varying density is given by the hydrostatic law: $dP = -\rho(z) g dz$.Integrating from the bottom ($z=0$, $P=P_1$) to the top ($z=H$, $P=P_2$):$\int_{P_1}^{P_2} dP = -\int_{0}^{H} \rho(z) g dz$$P_2 - P_1 = -g \int_{0}^{H} \rho_0 \left[ 2 - \left( \frac{z}{H} \right)^2 \right] dz$$P_1 - P_2 = g \rho_0 \int_{0}^{H} \left( 2 - \frac{z^2}{H^2} \right) dz$$P_1 - P_2 = g \rho_0 \left[ 2z - \frac{z^3}{3H^2} \right]_{0}^{H}$$P_1 - P_2 = g \rho_0 \left( 2H - \frac{H^3}{3H^2} \right)$$P_1 - P_2 = g \rho_0 \left( 2H - \frac{H}{3} \right)$$P_1 - P_2 = g \rho_0 \left( \frac{5H}{3} \right) = \frac{5}{3} \rho_0 g H$.

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