If 80 percent of radioactive element is left after 250 years, then what percent remains after 600 years? What is the half life of the element?
Assuming exponential decay
amount = a e^(kt) where a is the initial amount, and t is number of years
.8 = 1 e^(250k(
ln .8 = ln e^(250k)
ln .8 = 250k
k = ln.8/250
after 600 years ?
amount = 1 e^(600ln.8/250) = .5853 or appr 58.5%
for half-life ?
.5 = e^(ln.8 t/250)
ln.5 = ln.8/250 t
t = 250ln .5/ln .8 = 776.6 years
Well, if 80 percent of the radioactive element remains after 250 years, then after 600 years... drumroll please... probably some percentage less than 80 percent!
As for the half-life of the element, it's like the ultimate disappearing act. Imagine this radioactive element as your ex's phone number. Every time you look at it, half of those digits vanish into thin air. So, the half-life is the time it takes for half of the radioactive substance to decay.
Since we know that 80 percent is left after 250 years, we can infer that the half-life is longer than 250 years. Unfortunately, I don't have access to the exact half-life of the element. But hey, look at the bright side: there's plenty of time left for science to figure it out.
To find the percentage of the radioactive element that remains after 600 years, we can use the formula:
\( \% \text{ Remaining } = 100 \times (1 - (\frac{1}{2})^{\frac{\text{Time}}{\text{Half-Life}}}) \)
Given that 80 percent of the element is left after 250 years, we can substitute these values into the formula:
\( 80 = 100 \times (1 - (\frac{1}{2})^{\frac{250}{\text{Half-Life}}}) \)
To find the half-life, we can rearrange the equation as follows:
\( (\frac{1}{2})^{\frac{250}{\text{Half-Life}}} = 1 - \frac{80}{100} \)
Now, let's solve this equation step-by-step.
Step 1: Simplify the right-hand side of the equation:
\( (\frac{1}{2})^{\frac{250}{\text{Half-Life}}} = 1 - \frac{80}{100} \)
\( (\frac{1}{2})^{\frac{250}{\text{Half-Life}}} = 1 - 0.8 \)
\( (\frac{1}{2})^{\frac{250}{\text{Half-Life}}} = 0.2 \)
Step 2: Take the logarithm of both sides using the base 2:
\( \log_2((\frac{1}{2})^{\frac{250}{\text{Half-Life}}}) = \log_2(0.2) \)
Step 3: Apply the logarithm properties:
\( \frac{250}{\text{Half-Life}} \times \log_2(\frac{1}{2}) = \log_2(0.2) \)
Step 4: Simplify:
\( \frac{250}{\text{Half-Life}} \times (-1) = \log_2(0.2) \)
Step 5: Solve for the half-life:
\( -\frac{250}{\text{Half-Life}} = \log_2(0.2) \)
\( \frac{250}{\text{Half-Life}} = -\log_2(0.2) \)
\( \text{Half-Life} = \frac{250}{\log_2(0.2)} \)
Using a calculator, we can find that the value of the half-life is approximately 166.941 years.
Now, to determine the percentage of the element that remains after 600 years, we can use the formula:
\( \% \text{ Remaining } = 100 \times (1 - (\frac{1}{2})^{\frac{600}{\text{Half-Life}}}) \)
Substituting the values into the formula:
\( \% \text{ Remaining } = 100 \times (1 - (\frac{1}{2})^{\frac{600}{166.941}}) \)
Using a calculator, we find that the percentage remaining is approximately 45.55 percent after 600 years.
Therefore, approximately 45.55 percent of the radioactive element remains after 600 years, and the half-life of the element is approximately 166.941 years.
To determine the percent of the radioactive element that remains after 600 years, we can use the formula for exponential decay:
N(t) = N₀ * (1/2)^(t/h)
Where:
N(t) is the amount remaining after time t
N₀ 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) = N₀ * (1/2)^(600/573.907)
Since we are given that 80 percent (or 0.8) remains after 250 years, we can substitute N₀ = 0.8 into the equation:
N(600) = 0.8 * (1/2)^(600/573.907)
Calculating this with a calculator gives us N(600) ≈ 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.