Acertain 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
 
 There will be 89% of the isotope left in about:
 t = 583 years