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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 find out in how many years the substance will contain exactly 20 grams of Carbon-14, we need to set A equal to 20 in the given equation and solve for t. This yields: 20 = 25e^0.00012t First, we isolate the exponential part by dividing both sides of the equation by 25: 20/25 = e^0.00012t 0.8 = e^0.00012t Now, to get rid of the exponential, we can use the natural logarithm function, ln, which is the inverse of the exponential function. ln(0.8) = ln(e^0.00012t) Applying the property of logarithms that allows us to bring t out as a multiplier (ln(a^b) = bln(a)) results in: ln(0.8) = 0.00012t * ln(e) Noting that ln(e) equals 1, we simplify the equation to: ln(0.8) = 0.00012t Finally, solve for t: t = ln(0.8) / 0.00012 So, the number of years needed for the substance to contain exactly 20 grams of Carbon-14, expressed in term of a natural logarithm, is t = ln(0.8) / 0.00012.
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