Final answer:
To find the half-life of the radioactive goo, we calculated the number of times the sample halved from 168 grams to 5.25 grams, which is five half-lives. Dividing the total decay time of 225 minutes by the number of half-lives, we determined that the half-life of the radioactive goo is 45 minutes.
Step-by-step explanation:
To determine the half-life of the radioactive goo, we can use the concept of half-life, which is the time it takes for half of a radioactive substance to decay. For instance, if we begin with 1.00 gram of a radioactive element, after one half-life, only 0.500 grams would remain, and so forth. The pattern of decay continues such that after each additional half-life period, only half of the remaining substance exists.
In this problem, the initial mass of our radioactive goo is 168 grams, and after 225 minutes it has decayed to 5.25 grams. To find out how many half-lives have passed, we can divide the final amount (5.25 grams) by the initial amount (168 grams) and calculate the number of times the quantity has halved.
Here is how the calculation works:
- After 1 half-life: 168 / 2 = 84 grams
- After 2 half-lives: 84 / 2 = 42 grams
- After 3 half-lives: 42 / 2 = 21 grams
- After 4 half-lives: 21 / 2 = 10.5 grams
- After 5 half-lives: 10.5 / 2 = 5.25 grams
Therefore, five half-lives have occurred to get from 168 grams to 5.25 grams. Since the total time taken is 225 minutes, we can find the time for one half-life by dividing the total time by the number of half-lives.
Half-life time = Total time / Number of half-lives = 225 minutes / 5 = 45 minutes.
Thus, the half-life of the radioactive goo is 45 minutes.