Question

Initially 100 milligrams of a radioactive substance was present. After 9 hours the mass had decreased by 2%. If the rate of decay is proportional to the amount of the substance present at time t, determine the half-life of the radioactive substance. (Round your answer to one decimal place.)

hr?

Answer 1

308.8 hours

Explanation:

You want the half-life of a substance that decays by 2% in 9 hours.

Exponential function

The function that predicts the amount remaining can be written two ways:

y = x₀·(1/2)^(t/h) . . . . . . where t is the half-life

y = x₀·(1 -2%)^(t/9) . . . . based on 2% loss in 9 hours

Equating these expressions, we can find h, the half-life.

x₀·(0.5)^(t/h) = x₀·(0.98)^(t/9)

Dividing by x₀ and taking logarithms, we have ...

(t/h)·log(0.5) = (t/9)·log(0.98)

Solving for h gives ...

h = 9·log(0.5)/log(0.98) ≈ 308.8 . . . . . hours

The half-life of the substance is about 308.8 hours.

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