The half-life of a 1000 gram substance is 50 years. If it decays according to an exponential growth model, the amount of time it will take for only 100 grams of the substance to remain is

Answer 1

P = Oe^kt where P is the new amount, O is the the original amount, t is time, and k is the change rate, in this case decay.

500 = 1000e^50k t = 5o, O = 1000, P = 50. Solve for k.

.5 = e^50k Take the ln of both sides

ln.5 = 50k

(ln.5)/50 = k

Use the k to find the amount remaining in the new situation.

100 = 1000e^((ln.5)/50)t)

.1 = e^(((ln.5)/50))t)

ln.1 = ((ln.5)/50)t

ln.1/((ln.5)/50) = t = about 166.096 years