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SAT Subject Math Test Level 2
When a certain radioactive element decays, the amount at any time t can be calculated using the function
\[E(t) = ae^{t/500}\]
where a is the original amount and t is the elapsed time in years. How many years would it take for an initial amount of 250 milligrams of this element to decay to 100 milligrams?
(A) 125 years (B) 200 years (C) 458 years (D) 496 years (E) 552 years Solution
(C).
Plot the graphs of y1 = 250e^(t/500) and y2 = 100. Using the answer choices and information in the problem, find the point when y1 and y2 intersect.
From the graph, you will notice that the x intercept is between 400 and 500, so the answer is 458.
An alternative solution is to substitute 250 for a and 100 for E to get 100 = 250e^(t/500). Divide both sides by 250. Take the ln of both sides of the equation. Finally, multiply both sides by 500 to get t = 500 ln 2/5 =~ 458.
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