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Question: A scientist measures the initial amount of Carbon-14 in a substance to be 25 grams. The relationship between A , the amount of Carbon-14 remaining in that substance, in grams, and t , the elapsed time, in years, since the initial measurement is modeled by the following equation. A = 25e^0.00012t In how many years will the substance contain exactly 20 grams of Carbon-14 ? Give an exact answer expressed as a natural logarithm.
To solve for t, when A = 20 grams (you want to find out when the substance will contain exactly 20 grams of Carbon-14) in the equation A = 25e^0.00012t, you would first divide both sides by 25 to isolate e^0.00012t: 20/25 = e^0.00012t This simplifies to: 0.8 = e^0.00012t Then you would take the natural log of both sides: ln(0.8) = ln(e^0.00012t) By properties of logarithms, you can bring down the power (0.00012t) in front: ln(0.8) = 0.00012t * ln(e) Since ln(e) = 1, this simplifies further to: ln(0.8) = 0.00012t Finally, to solve for t, you would divide both sides by 0.00012: t = ln(0.8) / 0.00012 So, in exact terms, the substance will contain exactly 20 grams of Carbon-14 in ln(0.8) / 0.00012 years.
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