# A radioactive substance decays so that after t years, the amount remaining, expressed as a percent of the original amount, is A(t)=100(1.6)^(-t). Determine the function which represent the rate of decay of the substance, the half-life for this substance and the rate of decay when half the substance has decayed? When the radioactive substance decays the amount remaining after a time t is equal to A(t) = 100*(1.6)^(-t).

The rate at which the radioactive substance is decaying is the derivative of A(t) with respect to time,

dA(t)/dt = (-100*ln 1.6)*(1.6)^(-t)

The half life of the substance is the time after...

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When the radioactive substance decays the amount remaining after a time t is equal to A(t) = 100*(1.6)^(-t).

The rate at which the radioactive substance is decaying is the derivative of A(t) with respect to time,

dA(t)/dt = (-100*ln 1.6)*(1.6)^(-t)

The half life of the substance is the time after which 50% of the substance is left.

50 = 100*(1.6)^(-t)

=> 1/2 = (1.6)^(-t)

=> -t = log 0.5/log 1.6

=> t = 1.47 years

At the time when half the substance has decayed, the rate of decay is

(-100*ln 1.6)*(1.6)^(ln 0.5/ln 1.6)

=> -23.5 percent per year

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