It will take 527.09 years for the radioactive substance to reduce to 142 grams
The function is given as:
[tex]\mathbf{A = A_oe^{=0.0021t}}[/tex]
And the parameters are given as:
[tex]\mathbf{A = 142}[/tex] -- the current amount of the substance
[tex]\mathbf{A_o = 430}[/tex] --- the initial amount
So, the equation becomes
[tex]\mathbf{142= 430 \times e^{-0.0021t}}[/tex]
Divide both sides by 430
[tex]\mathbf{0.3302 = e^{-0.0021t}}[/tex]
Take logarithm of both sides
[tex]\mathbf{log(0.3302) = log(e^{-0.0021t})}[/tex]
This gives
[tex]\mathbf{log(0.3302) = log(0.9979^t})}[/tex]
Apply laws of logarithm
[tex]\mathbf{log(0.3302) = tlog(0.9979})}[/tex]
Make t the subject
[tex]\mathbf{t = \frac{log(0.3302)}{log(0.9979}}}[/tex]
[tex]\mathbf{t = 527.09}[/tex]
Hence, it will take 527.09 years to reduce to 142 grams
Read more about radioactive decays at:
https://brainly.com/question/1160651
