analysis of another rock shows that it contains 60% of its original element; the other 40% decayed into lead. How old is the rock
?
1st step: what is the half-life of the original element?
30%
To determine the age of the rock, we can use the concept of half-life, which is the time it takes for half of the original element to decay into another element. In this case, we know that 40% of the rock has decayed into lead, which means that 60% of the original element is still remaining.
Let's assume that the original element in the rock is called "X" and it decays into "lead" over time. Since we know that 40% of X has decayed into lead, it means that 60% of X is still remaining.
Now we need to find out how many half-lives it took for the amount of X to reduce to 60%. To do that, we can use the formula:
Remaining amount = Initial amount * (1/2)^(Number of half-lives)
In this case, the remaining amount is 60%, and the initial amount is 100%. Let's substitute the values:
60% = 100% * (1/2)^(Number of half-lives)
To isolate the number of half-lives, we can take the logarithm of both sides of the equation. Using base 2 logarithm (since it's a half-life), we have:
log2(60%) = log2(100% * (1/2)^(Number of half-lives))
log2(60%) = log2(100%) + Number of half-lives * log2(1/2)
log2(60%) = 0 + Number of half-lives * (-1)
Now we can solve for the number of half-lives by dividing both sides of the equation by -1:
Number of half-lives = log2(60%) / -1
Using a calculator, we find that log2(60%) is approximately -0.73697. Dividing this by -1, we get:
Number of half-lives = 0.73697
Since each half-life represents a fixed amount of time, we need to know the duration of one half-life to determine the rock's age. It depends on the specific element decaying, as each element has its own unique half-life. If you can provide the element, I can help you determine the age of the rock.