To determine the percent of the radioactive element that remains after 600 years, we can use the formula for exponential decay:
N(t) = Nโ * (1/2)^(t/h)
Where:
N(t) is the amount remaining after time t
Nโ is the initial amount
t is the time that has passed
h is the half-life of the element
We are given that 80 percent (or 0.8) of the element remains after 250 years. Let's use this information to find the half-life of the element.
0.8 = 1 * (1/2)^(250/h)
To solve for h, we can take the logarithm of both sides. Let's use the natural logarithm (ln):
ln(0.8) = ln(1) + ln(1/2)^(250/h)
ln(0.8) = 0 + (250/h) * ln(1/2)
Using a calculator, we can find that ln(0.8) is approximately -0.2231. Plugging this value in, we have:
-0.2231 = (250/h) * ln(1/2)
Now, let's solve for h:
h = (250/ln(1/2)) * -0.2231
Using a calculator, we find that h is approximately 573.907 years.
Now that we know the half-life of the element, we can find the percent remaining after 600 years:
N(600) = Nโ * (1/2)^(600/573.907)
Since we are given that 80 percent (or 0.8) remains after 250 years, we can substitute Nโ = 0.8 into the equation:
N(600) = 0.8 * (1/2)^(600/573.907)
Calculating this with a calculator gives us N(600) โ 0.4981, or approximately 49.81 percent.
Therefore, after 600 years, approximately 49.81 percent of the radioactive element remains. The half-life of the element is approximately 573.907 years.