Final answer:
To find the number of years it will take for the substance to contain exactly 20 grams of Carbon-14, we set up the equation 20 = 25e^(-0.00012t) and solve for t. The exact answer, expressed as a natural logarithm, is t = -ln(0.8) / 0.00012 years.
Step-by-step explanation:
The relationship between the amount of Carbon-14 remaining in a substance, A, and the elapsed time, t, since the initial measurement is given by the equation A = 25e^(-0.00012t), where e is the base of natural logarithms. We need to find the value of t when A is 20 grams.
To solve for t, we can set the equation equal to 20 and solve for t:
20 = 25e^(-0.00012t)
Divide both sides of the equation by 25:
0.8 = e^(-0.00012t)
To isolate t, take the natural logarithm of both sides of the equation:
ln(0.8) = -0.00012t
Finally, divide both sides of the equation by -0.00012 to solve for t:
t = -ln(0.8) / 0.00012
The exact answer to this equation, expressed as a natural logarithm, is t = -ln(0.8) / 0.00012 years.