You want to concentrate a slurry from $5 %$ (by vol.) solids to $30 %$ (by vol.) in a thickener. The solids density is $200 \mathrm{lb}_{\mathrm{m}} / \mathrm{ft}^{3}$ and that of the liquid is $62.4 \mathrm{lb}_{\mathrm{m}} / \mathrm{ft}^{3}$. A batch settling test was run on the slurry, and the analysis of the tests yielded the following information:

$\boldsymbol{\varphi}$ (Vol. Fraction Solids)	Settling Rate $\left(\mathbf{l b}_{\mathbf{m}} / \mathbf{h} \mathbf{f t}^{2}\right)$
0.05	73.6
0.075	82.6
0.1	79.8
0.125	70.7
0.15	66
0.2	78
0.25	120
0.3	200

If the feed flow rate of the slurry is $500 \mathrm{gpm}$, what should the cross-sectional area of the thickener tank be? What are the overflow and underflow rates?