The time needed for a 400-grams radioactive element with half-life of 13 minutes to decay to 4 grams is 86.4 minutes.
The half-life formula of a decay system is given by:
A = Ao x (0.5)^(t/h)
Where:
A = final amount
Ao = Initial amount
t = time period
h = half-life
Parameters given in the problem:
h = 13 minutes
Ao = 400 grams
A = 4 grams
Plug these parameters into the formula:
4 = 400 x (0.5)^(t/13)
(0.5)^(t/13) = 0.01
t/13 x log(0.5) = log (0.01)
t/13 = 6.64
t = 6.64 x 13 = 86.4 minutes
Hence, it takes 86.4 minutes for the element X to decay to 4 grams.
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