Final answer:
To find the number of half-lives, we use the equation relating the amount remaining to the initial amount and the number of half-lives. Four half-lives have passed, and thus the total time of decay is 16.168 minutes.
Step-by-step explanation:
To determine how many half-lives will have passed before a substance decays from 115.2 grams to 7.2 grams, we use the relation amount remaining = initial amount × (1/2)^n, where n is the number of half-lives. In this example, the initial amount is 115.2 grams and the amount remaining is 7.2 grams.
Setting up the equation, we get 7.2 = 115.2 × (1/2)^n.
Dividing both sides by 115.2 gives us (1/2)^n = 7.2/115.2
= 1/16.
Since (1/2)^4 = 1/16, we know that n = 4.
Therefore, 4 half-lives have passed.
To calculate the total time of decay, we multiply the number of half-lives by the half-life of the substance:
4 × 4.042 minutes = 16.168 minutes.