An air sample is contaminated with ¹⁵O,which has a half-life of 2.03 minutes.You can pass it through a long pipe to allow it to decay until it can be safely released,at an air speed of 1.1 m/s.How long must your pipeline be for the sample to decay to 3% of its original activity?

Question

Quizplus · Accepted Answer

We can use the radioactive decay formula to solve the problem:
A(t) = A0 * (1/2)^(t/T)
where:
A(t) = activity at time t
A0 = initial activity
t = time
T = half-life

We want to find the time it takes for the activity to decay to 3% of the initial activity, so we can write:
0.03*A0 = A0 * (1/2)^(t/T)
Taking the natural logarithm of both sides:
ln(0.03) = ln(1/2)^(t/T)
Using the rule that ln(a^b) = b*ln(a):
ln(0.03) = (t/T)*ln(1/2)
Solving for t:
t = (T/ln(1/2)) * ln(0.03)

Substituting T = 2.03 minutes, we get:
t = (2.03/ln(1/2)) * ln(0.03) ≈ 16.5 minutes

During this time, the contaminated air travels at a speed of 1.1 m/s, so the length of the pipeline needs to be:
length = speed * time = 1.1 m/s * 16.5 min * 60 s/min ≈ 678 m

Therefore, the best choice is A) 678 m.

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