An unknown radioactive element decays into non-radioactive substances. in 180 days the radioactivity of a sample decreases by 73 percent.
(a) what is the half-life of the element? (in days)
(b) how long will it take for a sample of 100 mg to decay to 60 mg? (in days)

Step-by-step explanation:

a) The proportion remaining (p) after d days can be described by ...

  p = (1 -0.73)^(d/180) = 0.27^(d/180)

Then p=1/2 when ...

  0.50 = 0.27^(d/180)

  log(0.50) = (d/180)log(0.27)

  180(log(0.50)/log(0.27) = d ≈ 95.3

The half-life is about 95.3 days.

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b) For the proportion remaining to be 60/100, we can use the same solution process. In the end, 0.50 will be replaced by 0.60, and we have ...

  d = 180(log(0.60)/log(0.27) ≈ 70.2 . . . days

60 mg will remain of a 100 mg sample after 70.2 days.

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