A radioactive material disintegrates at a rate proportional to the amount curren
ID: 2972528 • Letter: A
Question
A radioactive material disintegrates at a rate proportional to the amount currently present. If Q(t) is the amount present at time t, then dQ/dt= -rQ where r > 0 is the decay rate. (a) If 100 mg of a mystery substance decays to 80.04 mg in 1 week, determine the decay rate r. Round the answer to 5 decimal places. (b)Find an expression for the amount of this substance present at any time t. Use the value of r found in the previous step. (c)Find the time required for the substance to decay to one-half its original amount. Round the answer to 3 decimal places.Explanation / Answer
dQ/dt = -rQ dQ/Q = -r dt Integrate both sides ? dQ/Q = ? -r dt ln(Q) = -rt + C0 Q = e^(-rt+C0) Q = e^(-rt) * e^(C0) Q = C e^(-rt) Q(0) = 100 C e^0 = 100 C = 100 Q(t) = 100 e^(-rt) Now is where things get confusing. Is t measured in days or weeks? Problem does not say. If t is measured in weeks, then we get: Q(1) = 82.04 100 e^(-r) = 82.04 e^(-r) = 0.8204 -r = ln(0.8204) r = -ln(0.8204) ˜ 0.19796325 If t is measured in days, then we get: 100 e^(-7r) = 82.04 e^(-7r) = 0.8204 -7r = ln(0.8204) r = -ln(0.8204)/7 ˜ 0.02828046 PLEASE RATE ME