= Tapered microplate optical path model
A \b[<tapered microplate optical path model>] accounts for a well's changing horizontal area when relating liquid volume to the vertical light path. The ideal <microplate fill-height optical cancellation> is exact for parallel sides; a tapered well generally breaks that cancellation even for a fully dissolved, homogeneous dye.
Let the internal area at height $z$ be $S(z)$ and the filled height be $h$. For dissolved amount $n$ distributed uniformly in volume $V(h)$,
$$
V(h)=\int_0^h S(z)\,\mathrm dz,\qquad
c=\frac{n}{V(h)},\qquad
B\equiv A-A_0=\epsilon h c=\frac{\epsilon n h}{V(h)}.
$$
The last equality uses the <Beer-Lambert law> for an axial beam that traverses the whole liquid height. Thus the relevant area is the height-averaged area $V(h)/h$, not automatically the bottom area or the area at the meniscus.
For an original area-linear model $S(z)=S_0(1+\alpha z)$, with positive area throughout the liquid,
$$
V(h)=S_0h(1+\alpha h/2),\qquad
B=\frac{\epsilon n}{S_0(1+\alpha h/2)}.
$$
For outward taper $\alpha>0$, raising the fill height at fixed dissolved amount lowers the ideal <absorbance>. With stipulated $S_0=0.50\,\mathrm{cm^2}$ and $\alpha=1\,\mathrm{cm^{-1}}$, heights $0.20$ and $0.40\,\mathrm{cm}$ correspond to volumes $0.110$ and $0.240\,\mathrm{cm^3}$. The signal ratio is $1.10/1.20=11/12$, despite unchanged product amount. These are geometric examples, not dimensions assigned to a commercial plate.
Area increasing linearly with height is not an exact conical frustum. If the radius instead follows $r(z)=r_0+\beta z$, integration gives
$$
V(h)=\pi\left(r_0^2h+r_0\beta h^2+\frac{\beta^2h^3}{3}\right),\qquad
B=\frac{\epsilon n}{\pi(r_0^2+r_0\beta h+\beta^2h^2/3)}.
$$
Both models recover the constant-area limit when taper vanishes. They assume the same <molar absorption coefficient> and full product recovery at every volume. The manufacturer separates <MTT> product formation from dissolution and optical measurement.<Cell Proliferation Kit I MTT instructions by Roche>[Section3.1 and Figure4, printed p.9]{ref} Suspended particles, a curved meniscus, an oblique beam or a reader's <microplate path-length correction> need additional treatment. A change in signal with dilution therefore does not, by itself, establish a change in <formazan> amount or <cell viability>.
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