When radioactive substances decay, the amount remaining will form a geometric sequence when measured over constant intervals of time. The table shows the amount of a radioactive isotope initially and after 2 hours. What are the amounts left after 1 hour, 3 hours, and 4 hours?
Hours Elapsed
0
1
2
3
4
Grams of the isotope
1284
326
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Part 1
How much of the isotope is left after 1 hour?
To find the amount of the isotope left after 1 hour, we need to determine the common ratio between each subsequent time interval.
Initial amount = 1284 grams
Amount after 2 hours = 326 grams
Common ratio = (Amount after 2 hours) / (Initial amount) = 326 / 1284 = 0.2537
Amount after 1 hour = Initial amount * common ratio = 1284 * 0.2537 ≈ 325.5 grams
Therefore, after 1 hour, there are approximately 325.5 grams of the isotope left.
Part 2
How much of the isotope is left after 3 hours and 4 hours?
Amount after 3 hours = Amount after 2 hours * common ratio = 326 * 0.2537 ≈ 82.7 grams
Amount after 4 hours = Amount after 3 hours * common ratio = 82.7 * 0.2537 ≈ 21 grams
Therefore, after 3 hours, there are approximately 82.7 grams of the isotope left, and after 4 hours, there are approximately 21 grams left.