The half-life of polonium is 139 days, but your sample will not be useful to you after 95 % of the
radioactive nuclei present on the day the sample arrives has disintegrated. For about how many days
after the sample arrives will you be able to use the polonium?
you want to know when you have 5% left.
Let P = c(2)^(-t/13), where c is the amount you started with, and t is the number of days.
so we want .05c=c(2)^(-t/139)
I used logs to solve and got t = 600 days
t=139ln(.05)/(ln(1/2))
t=139ln(.05)/(ln(1)-ln(2))
t=139ln(.05)/(0-ln(2))
t=139ln(.05)/(-ln(2))=600.7480052=600 days
t=(half life* ln(percent of sample left))/(ln(1/2))
To find out how many days you will be able to use the polonium after the sample arrives, you can use the equation:
0.05c = c * 2^(-t/139)
Simplifying this equation, you can cancel out the c:
0.05 = 2^(-t/139)
To solve for t, you can take the logarithm of both sides of the equation:
log(0.05) = log(2^(-t/139))
Using the property of logarithms, you can rewrite the equation as:
log(0.05) = (-t/139) * log(2)
Solving for t, you can rearrange the equation:
t = -139 * (log(0.05) / log(2))
Using a calculator, you can calculate the value of t:
t ≈ -139 * (-2.32193 / 0.69315)
t ≈ 600
Therefore, you will be able to use the polonium for approximately 600 days after the sample arrives.
To solve for the number of days after the sample arrives that you will be able to use the polonium, we can set up an equation using the concept of exponential decay.
Let P be the amount of polonium present at any given time, c be the initial amount of polonium when the sample arrives, and t be the number of days.
We know that the half-life of polonium is 139 days, which means that half of the radioactive nuclei will have disintegrated every 139 days. Therefore, the amount of polonium remaining after t days can be represented as P = c(2)^(-t/139).
Now, we want to find t when the remaining polonium is 5% of the initial amount, which means we have 0.05c left.
Setting up the equation, we have:
0.05c = c(2)^(-t/139)
To solve for t, we can divide both sides of the equation by c:
0.05 = (2)^(-t/139)
To eliminate the exponent, we can take the logarithm (base 2) of both sides:
log2(0.05) = log2((2)^(-t/139))
Using the property of logarithms, we can bring down the exponent:
log2(0.05) = -t/139 * log2(2)
Since log2(2) equals 1, we have:
log2(0.05) = -t/139
Now, we can solve for t by isolating it on one side:
-t/139 = log2(0.05)
Multiplying both sides by -139:
t = -139 * log2(0.05)
Using a calculator, we find that log2(0.05) is approximately -4.321 and evaluating the expression:
t ≈ -139 * (-4.321)
t ≈ 598.2
Rounding to the nearest whole number, we can conclude that you will be able to use the polonium for about 600 days after the sample arrives.