Differential equation describing the radio active decay of |3| iodine is given b

Question

Differential equation describing the radio active decay of |3| iodine is given by dR / dt = -0.084 R R(0) = 20 Where t is days. Solve this differential equation and find the half life for iodine R(t) = _______ Half life = _______ Suppose a technician is receiving exposure in mCi/day.The total exposure over 14 days is given by the integral P(t) = 4e-0.084t D(t) dt Find this total exposure total Exposure = _________ In general the exposure time T is given by the expression P(t) dt = A Where A is the total exposure or radiation dosage. How long can the tech stay near this source if the exposure is to be kept to less than A= 10 mCi ? Exposure time T = ________

Explanation / Answer

dR/dt = -0.084R

dR /R = -0.084dt

integral(dR/R) = -0.084integral(dt)

lnR = -0.084t +C

R(t) = e^(-0.084t) * C

at t=0,R=20

20 = e^(0) C

C=20

R(t) = 20e^(-0.084t)

D(t) = 4e^(-0.084t)

integral(0 to 14) {D(t) dt} = integral(0 to 14) {4e^(-0.084t) dt}

= 4e^(-0.084t) /(-0.084) |(0 to 14)

= -4/0.084 {e^(-0.084*14) - e^0}

=-4/0.084 {e^(-1.176) - 1} = 47.6 {e^(-1.176) - 1}

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