A certain radioactive isotope decays at a rate of 2% per 100 years. If t represents the amount of the isotope left then the equation for the situation is

y=y0e^-0.0002t

A certain isotope decays at a rate of 2%per 100 years. If t represent the time in years and y represent the amount of the isotope left then the equation for the situation is y=y0e^-0.0002t. In how many years will there be 86% of the isotope left?

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Let x = initial amount

then our formula is

.86x = xe^ (-0.0002t)

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Notice, if we divide both sides by x, we eliminate our unknown:

.86 = e^ (-0.0002t)

Solving for x, we take the ln of both sides:

ln (.86) = - 0.0002t

ln (.86) / (-0.0002) = t

754.114 years = t