It will take approximately 28 years to decay 15 gram sample to 12 gram.
Given,
The yearly decaying of Plutonium-238 = 7.9 × 10⁻³
The mass of original sample = 15 grams
We have to calculate the time taken to decay to 12 grams.
Here,
At = A0 × e^(-k × t)
At = 12 g
A0 = 15 g
k = 7.9 × 10⁻³ = 0.0079
We have to find t.
That is,
At = A0 × e^(-k × t)
12 = 15 × e^(-0.0079 × t)
12/15 = e^(-0.0079 × t)
0.8 = e^(-0.0079 × t)
Now,
Logarithm both sides (because ln(e) = 1:
ln(0.8) = ln(e^(-0.0079 × t))
ln(0.8) = (-0.0079 × t) × ln(e)
-0.223 = -0.0079 × t
t = -0.223 / -0.0079
t = 28.23
t ≈ 28 years
Therefore,
It will take approximately 28 years to decay 15 gram sample to 12 gram.
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