Complete exponential model by solving for k?

Question

I have a big finals test tomorrow and I've studied everything except this one problem. I've been stuck on this problem for about an hour now, and I've looked through notes, the textbook, and other online sources and got nothing. So if you would take it out of your time to answer this, I'd really be greatful. I already know the answer, but i want to know the process of how you get there and what equation you use. So please don't post just answers. Thanks.

A piece of fossil contains 15% of its original amount of carbon. How old is the fossil if the half-life of carbon is 5715 years? (Hint: Complete the exponential model by solving for k. Dont forget to plug in k at the end and write y=..., then use the model to date the fossil.)

Traveler · Accepted Answer

Assuming 100 grams as a starting point, you'd have 50 grams after 5715 years. 
Use this relationship initially:

50 =100e^(k*5715)
50/100 = e^(k*5715)

Take natural log of both sides:
ln 0.5 = k*5715
-0.69315 = k*5715
-0.00012128 = k (which is -0.012128%)

Now you know "k" and can figure out the amount of time to get 15% of the original amount:
15 = 100e^(-0.00012128*t)
15/100 = e^(-0.00012128*t)

Take the natural log of both sides:
ln (0.15) = -0.00012128*t
-1.897119 = -0.00012128*t
15642.5 years = t

So generically for any amount of carbon that existed initially, you'd have:

y = (Initial amount of Carbon) * e^(-0.00012128*t)

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Complete exponential model by solving for k?

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