A fossilized leaf is analyzed. Unlike standard carbon-14 dating which assumes constant initial concentration $C_0$, the local atmospheric $C^{14}$ level at the time of the leaf's death $t$ years ago is modeled by $C_{init}(t) = C_0 (1 + 0.1 \\sin(\\omega t))$, where $C_0 = 100$ ppm, $\\omega = 0.001$ year$^{-1}$, and the decay constant for $C^{14}$ is $\\lambda = 1.2 \\times 10^{-4}$ year$^{-1}$. If the current $C^{14}$ concentration of the fossil is $40$ ppm, which transcendental equation must be solved to find its age $t$?