Assume the substance has a half-life of 11 years and the initial amount is 126 grams. How long will it be until only 15% remains?

(126) times (1/2 to the x/11 power) = 15% times 126
(1/2) to the x/11 power = 0.15
Take the log of both sides :
(x/11) times log(1/2) = log(0.15)
Multiply both sides by 11 :
'x' times log(1/2) = 11 x log(0.15)
Divide both sides by " log(1/2) " :
x = 11 x log(0.15)/log(1/2) = 30.107 years (rounded)
That's the time it takes for any-size sample of this substance to decay to 15% of its original size.

Answered by AL2006 | 2024-06-10

Calculating Remaining Radioactive Substance
To determine how long it will take for only 15% of a substance with a half-life of 11 years to remain, we can use the half-life formula which is:
N(t) = N 0 ​ . ( 1/2 ) ( t / T )
Where:

N(t) is the amount of substance remaining after time t
N_0 is the initial amount of the substance
t is the time that has passed
T is the half-life of the substance

First, we determine what fraction of the substance 15% is:
15% = 15/100 = 0.15
We want the remaining amount of the substance (N(t)) to be 15% of the initial amount (N_0), so we set up the equation:
0.15 · 126g = 126g · ( 1/2 ) ( t /11 ye a rs )
Now, solve for t:
0.15 = ( 1/2 ) ( t /11 )
To solve for t, take the logarithm of both sides, then solve for t:
log(0.15) = (t/11) · log(1/2)
t = 11 · (log(0.15) / log(1/2))
Calculate t using a calculator:
t ≈ 11 · (log(0.15) / log(0.5))
t ≈ 11 · (-0.8239 / -0.3010)
t ≈ 11 · 2.7359
t ≈ 30.095 years
Therefore, it will take approximately 30 years for 15% of the substance to remain.

Answered by Anastasia2022 | 2024-06-24

It will take approximately 30.1 years for the substance to decay to 15% of its initial amount. This is calculated using its half-life of 11 years and logarithmic decay equations. By determining the fraction that remains and applying the appropriate decay formula, we find the time required for the desired amount.
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Answered by AL2006 | 2024-12-24