Sodium-24 has a half-life of approximately 15 hours. If only one-eighth of the sodium-24 remains, about how much time has passed?

Question

a
15 hours
b
45 hours
c
30 hours
d
60 hourss

Answer 1

The correct answer is b) 45 hours.

To solve this problem, we can use the formula for exponential decay:

Amount remaining = (initial amount) * (1/2)^(time elapsed / half-life)

We are given that only one-eighth of the sodium-24 remains, so the amount remaining is 1/8 of the initial amount. Therefore, we can set up the following equation:

1/8 = (1) * (1/2)^(time elapsed / 15)

To solve for the time elapsed, we can take the logarithm of both sides of the equation:

log(1/8) = log((1/2)^(time elapsed / 15))

Simplifying the equation, we get:

-3 = (time elapsed / 15) * log(1/2)

Now we can solve for the time elapsed:

time elapsed / 15 = -3 / log(1/2)

time elapsed = -3 * 15 / log(1/2)

Using a calculator, we find that log(1/2) is approximately -0.30103. Therefore:

time elapsed ≈ -3 * 15 / (-0.30103) ≈ 45 hours

So about 45 hours has passed.