Question #21959

75% of a sample was decayed in 24 hours , find the half life of the sample.

Expert's answer

Let X0 be the initial amount of the sample.

Assume that the amount of sample at time t is given bythe formula

X(t) = X0 e^(-at)

for some constant a>0.

We should find T such that

X(T)/X0 = e^(-aT) = 0.5

For this we need to know value of a.

By assumption at time t=24 hours

0.75 = X(24)/X0= e^(-a*24),

whence

a = -ln(0.75) /24

= 0.011987

Thus

X(t) = X0e^(-0.011987 t).

Hence fior half life T we have that

e^(-0.011987 *T) = 0.5

and so

T =-ln(0.5)/0.011987

= 57.825hours

Thus the half life of the sample is 57.825 hours.

Assume that the amount of sample at time t is given bythe formula

X(t) = X0 e^(-at)

for some constant a>0.

We should find T such that

X(T)/X0 = e^(-aT) = 0.5

For this we need to know value of a.

By assumption at time t=24 hours

0.75 = X(24)/X0= e^(-a*24),

whence

a = -ln(0.75) /24

= 0.011987

Thus

X(t) = X0e^(-0.011987 t).

Hence fior half life T we have that

e^(-0.011987 *T) = 0.5

and so

T =-ln(0.5)/0.011987

= 57.825hours

Thus the half life of the sample is 57.825 hours.

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