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Hint

I(0) is proportional to Σ_iΔSLD_i² V_i²,

where ΔSLD_i = SLD_i - SLD_solvent is the contrast (or excess scattering length density) of subunit i, and V_i is the volume of subunit i (the subunits of a core-shell particles are the the core and the shell).

In this problem, ΔSLD_core = -ΔSLD_shell, so the volumes of the core and shell must be equal to get I(0)=0. Volume of the shell: 4π(R_outer³-R_core³)/3

Set this equal to the volume of the core, 4πR_core^{3, and find a relation between the radii: ratio = R_outer/R_core=?}