Tank fills until the orifice outflow $C_dA_o\sqrt{2gh}$ balances the constant inflow $Q_{in}$; the balance level is the equilibrium level.
Find. The equilibrium level $h$ for gasoline and for oil; compare, and discuss the effect of raising the temperature to 40°C.
Approach. At equilibrium the orifice discharge (Torricelli's law scaled by $C_d$) exactly balances the constant inflow, $Q_{in}=C_dA_o\sqrt{2gh}$; solve for $h$ for each fluid's $C_d$ and compare against the 500 mm tank height.
Part (a) — gasoline equilibrium level. Orifice area $A_o=\tfrac{\pi}{4}(0.010)^2=7.854\times10^{-5}\ \text{m}^2$. Solving $Q_{in}=C_dA_o\sqrt{2gh}$ for $h$,
$$h_{gas}=\frac{1}{2g}\left(\frac{Q_{in}}{C_dA_o}\right)^2=\frac{1}{2(9.81)}\left(\frac{2.0\times10^{-4}}{(0.90)(7.854\times10^{-5})}\right)^2=\boxed{0.408\ \text{m} = 408\ \text{mm}}$$
which is below the 500 mm rim, so gasoline reaches a genuine steady equilibrium.
Part (b) — oil equilibrium level. With $C_d=0.75$,
$$h_{oil}=\frac{1}{2(9.81)}\left(\frac{2.0\times10^{-4}}{(0.75)(7.854\times10^{-5})}\right)^2=\boxed{0.588\ \text{m} = 588\ \text{mm}}$$
which EXCEEDS the 500 mm tank height. Checking the outflow at the rim itself confirms this: at $h=0.500$ m the oil orifice can only pass $C_dA_o\sqrt{2gh}=0.75(7.854\times10^{-5})\sqrt{2(9.81)(0.5)}=1.845\times10^{-4}\ \text{m}^3/\text{s}=0.1845\ \text{L/s}$, which is less than the 0.20 L/s inflow, so $\boxed{\text{oil never reaches equilibrium — the tank overflows}}$.
Part (c) — comparison. Gasoline settles to a stable 408 mm level, comfortably inside the tank, whereas oil's lower discharge coefficient (0.75 vs 0.90, a consequence of its much higher viscosity giving a lower orifice Reynolds number and more velocity-profile contraction/friction loss at the vena contracta) means even a completely full 500 mm tank cannot pass the required 0.20 L/s — the oil case overflows continuously rather than settling.
Part (d) — effect of raising the temperature to 40°C. Both fluids' viscosities fall with rising temperature (SAE 30 oil far more steeply than gasoline, per the Absolute-Viscosity attachment), which raises the orifice Reynolds number and pushes $C_d$ upward toward its high-Re asymptote for both fluids — more so for the oil, whose $C_d$ is furthest from that asymptote at 20°C. The parameter that changes is therefore $C_d$ itself (via viscosity/Reynolds number), not the driving head relation. Quantitatively, the oil would need $C_d$ to rise only to
$$C_{d,need}=\frac{Q_{in}}{A_o\sqrt{2gH_{tank}}}=\frac{2.0\times10^{-4}}{(7.854\times10^{-5})\sqrt{2(9.81)(0.5)}}=\boxed{0.813}$$
(an 8.4% rise from 0.75) to just avoid overflowing at the full 500 mm level, which is a plausible outcome of the substantial viscosity drop SAE 30 oil undergoes between 20°C and 40°C; gasoline's $C_d$ is already close to its high-Re ceiling, so its equilibrium level would drop only slightly below 408 mm.
