Radioactive polonium has a half-life of 2183.05 minutes. How many hours will it take for 70% of a sample to decay (leaving 30% of the original sample)? Give the answer rounded to the nearest tenth of an hour.
Let’s start by looking at a general form of an exponential function and figuring out what the various parts mean:
y = a(b)^x
The a and b are constants that depend on the particular problem. The x and y are the variables, and x is the exponent of b; if we put in a value for x, we can calculate a value for y (assuming we already know what the constants, a and b, are).
In “half-life” problems, we’re using the above equation. The “a” constant is our starting amount. “b” is set to ½.
The “x” is the “amount of time we’re interested in” divided by the half-life period. Dividing the elapsed time by the half-life tells us the number of half-lives that have gone by.
The “y” is the final amount after a given time period has elapsed:
Final amount = (initial amount)(1/2)^[(elapsed time)/(half-life)]
In your specific problem, we’re given the all the information on the initial amount, half-life, and final amount:
.3 = (1.00)(1/2)^(t/2183.05)
Solve for t. (Hint: use logarithms.) The answer will be in minutes. Convert to hours.
