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?

Question

Hours Elapsed
0
1
2
3
4
Grams of the isotope
1284
326
Question content area bottom
Part 1
How much of the isotope is left after 1​ hour?

Answer 1

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.