Iron -59 has a half life of 45.1 days. How old is an iron nail if the Fe-59 content is 25% that a new sample of iron? Show all calculations leading to a solution. Can you work the problem? I don't know how to solve for t!!! Please.

I don't intend to work this problem for you under any name. You take those numbers I gave you earlier and substitute them. Telling me you don't know how to solve for t doesn't help. Substitute the numbers, show me your work, go as far as you can, then tell me what you don't understand about the next step. That's the only way I will know what you don't understand.

It's half life - try looking in your book.

Sure! I'd be happy to help you with the calculations.

To find the age of the iron nail, we can use the concept of half-life and the given information.

The half-life of Iron-59 (Fe-59) is 45.1 days. This means that after every 45.1 days, half of the Fe-59 atoms decay or transform into another element. In this case, we are given that the Fe-59 content is 25% of that in a new sample of iron.

To solve for the age of the iron nail (t), we can use the following formula:

(Fe-59 Final Amount) = (Fe-59 Initial Amount) * (0.5)^(t / half-life)

In this formula:
- (Fe-59 Final Amount) represents the current amount of Fe-59 in the iron nail.
- (Fe-59 Initial Amount) represents the initial amount of Fe-59 in the iron nail (which was 100% in the new sample).
- t represents the age of the iron nail in days.
- half-life represents the half-life of Fe-59, which is 45.1 days.

Since we know that the Fe-59 content is currently 25% of the initial amount, we can substitute these values into the formula:

0.25 = 1 * (0.5)^(t / 45.1)

To solve for t, we need to isolate the variable t. We can start by taking the natural logarithm (ln) of both sides of the equation to eliminate the exponential:

ln(0.25) = ln((0.5)^(t / 45.1))

Using logarithm properties, we can simplify the equation further:

ln(0.25) = (t / 45.1) * ln(0.5)

Now, we can solve for t by multiplying both sides of the equation by (45.1 / ln(0.5)):

t = (45.1 / ln(0.5)) * ln(0.25)

Using a calculator, we can find ln(0.25) and ln(0.5) to substitute in the formula:

t ≈ (45.1 / -0.693) * -1.386

Simplifying further:

t ≈ 31.8 days

Therefore, the age of the iron nail is approximately 31.8 days.