A certain radioactive isotope decays at a rate of 2% per 100 years. if t represents time in years and y represents the amount of the isotope left then the equation for the situation is y=y0e-0.0002t. in how many years will there be 89% of the isotope left? round to the nearest year.

We rewrite the equation:

y = y0 * e ^ ( - 0.0002 * t)

In this equation:

I = represents the initial amount of the isotope

We have then:

89% of the isotope left:

0.89 * y0 = y0 * e ^ ( - 0.0002 * t)

We clear the time:

e ^ ( - 0.0002 * t) = 0.89

Ln (e ^ ( - 0.0002 * t)) = Ln (0.89)

-0.0002 * t = Ln (0.89)

t = Ln (0.89) / ( - 0.0002)

t = 582.6690813

round to the nearest year:

t = 583 years

Answer:

There will be 89% of the isotope left in а bout:

t = 583 years