Question

The rather dangerous element Wonderium decays radioactively to the beautiful metal Galgadotium at a rate equal to α times the amount of Wonderium present at time t, W(t). But Galgadotium decays at a rate equal to β times the amount of Galgadotium present at time t, G(t); here β is strictly larger than α. Give equations for W(t) and G(t), solve these equations, and use the solutions to compute limt→∞[W(t)/G(t)] in terms of αandβ ONLY, assuming that W(0)≠0. The correct answer is

Answer 1

Let wt be the amount of wonderium at timet and Go be the amount of Galga dotium at timet Since rate of delay of Wonderium is a times the amount of Wonderium present at time t. dwt - t & WH) - de regesents the Galgadotium decays at à rate & times the amount of Galgadotium at time to d GC +ß GO) - (2) VE 0 and ② are the equation for Wit and get from 0 dwe =+2 de Integrating from a tot wt) kawt - ta fit J. We ln was lost atat 7 lm We atat Am Woo La 27 WA = wo) etato This is the required Solution for wes, where Wo is the initial amount of Wonderium.

from (2) not to de Git) Integrating from otot ptdG9C4) - testet Jo Go de ln Alt a tot a) In Gt - +Bt 28 AM) = G Ca) etat -- This is the required solution for all, where Go is the initial amount of Galgadotium. Now, pyl eie fal a& X-B TO lim Wall = lim Woo etat finds Gift) - Glo et Bt I firm wo (-8) + = x 60) & B-ot lo - 0. as tena