An isotope has a half-life of 5,000 years, How long will it take to decay to 15% of its original quantity?

Question

rogue · Answer

$$f(t) = Ae ^{kt}$$$$k = \frac {-ln 2}{t _{\frac{1}{2}}} = \frac {-ln 2}{5000} = 0.000138629436 $$We'll let A = 100, so its easy to see the percent change.$$f(t) = 100e^{0.000138629436t} = 15$$$$.15 = e^{-0.000138629436t}$$$$t = \frac {\ln .15}{-0.000138629436} \approx 13685 years$$

rogue · Answer

The constant is negative in the exponential decay equation, I made a mistake when I first wrote the equation.
$$f(t) = Ae^{-kt}$$

anonymous · Answer

okay thank you. may i ask though how you found k?

rogue · Answer

Exponential decay is first order, so there is a formula that relates the half life and the constant.$$t _{\frac{1}{2}}^{} = \frac {-\ln 2}{k}$$

anonymous · Answer

okay, thank you very much!

rogue · Answer

Alright, no problem, good luck with your studies :)