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.

If 300 mg of a mystery substance decays to 80.84 mg in 3 weeks, find the time required for the substance to decay to one-half its original amount. Round the answer to 3 decimal places.

Answer 1

A radioactive material disintegrates at a rate proportional to the amount currently present. If Q(t) is the amount present at time t, then

Step 1)

First we will find the solution of the differential equaton

Integrate on both the side we can say that,

take anti log on both the side we can say that,

we know that e^ln(a) = a hence we can say that,

take e^C = Q₀ hence,

we can write,

--------------------------------------------------------------1)

Step 2)

As given If 300 mg of a mystery substance decays to 80.84 mg in 3 weeks

Hence put Q = 80.84, Q₀ = 300 and t = 3 in equation 1) and find the value of r

take ln on both the side we can say that,

we know that ln(e) = 1 hence we can say that,

put r = 0.43710 in equation 1) we can say that,

--------------------------------------------------2)

Step 3)

we have to find time required for the substance to decay to one-half its original amount

Hence put Q = 150, Q₀ = 300 in equation 2) and find the value of t

take ln on both the side we can say that,

we know that ln(e) = 1 hence we can say that,

rounded to three decimal places we can say that,

Hence we can say that it will take t = 1.586 weeks for the substance to decay to one-half its original amount