Problem 17 - College Algebra Essentials

Question

The half-life of the radioactive element krypton-91 is 10 seconds. If 16 grams of krypton-91 are initially present, how many grams are present after 10 seconds? 20 seconds? 30 seconds? 40 seconds? 50 seconds?

Step-by-Step Solution

Answer

After 10 seconds, 8 grams of Krypton-91 remain. After 20 seconds, 4 grams remain. After 30 seconds, 2 grams remain. After 40 seconds, 1 gram remains. After 50 seconds, 0.5 grams remain.

Step 1: Understand the Half-life Information

Given that the half-life of Krypton-91 is 10 seconds, this means that after each 10-second interval, the quantity of Krypton-91 will halve.

Step 2: Figure out How Many Half-Life Periods Have Passed

To figure out the quantity of Krypton-91 remaining after a certain number of seconds, compute the number of half-life periods that have passed. For example, at 10 seconds (half-life of Krypton-91), one half-life period has passed. At 20 seconds, two half-life periods have passed, and so forth.

Step 3: Calculate Quantity of Krypton-91 Remaining

The quantity of Krypton-91 remaining can be calculated by halving the initial quantity of Krypton-91 for each half-life period that has passed. Here's how: At 10 seconds (1 half-life), half of the initial 16 grams remain, which is 8 grams. At 20 seconds (2 half-lives), half of the remaining 8 grams is left, which is 4 grams. At 30 seconds (3 half-lives), half the remaining 4 grams is left, which is 2 grams. At 40 seconds (4 half-lives), half the remaining 2 grams is left, which is 1 gram. At 50 seconds (5 half-lives), half the remaining 1 gram is left, which is 0.5 grams.