In 1976 a research hospital bought half a gram of radium for cancer research. Assuming the hospital is still exists, how much of this radium will the hospital have in the year 6836 if the half-life of radium is 1620 years?
This this is a "half-life" situation we can use (1/2) as a base
amount = .5(1/2)^(t/1620)
if 1976--- t = 0, then 6836 ---> t = 4860
amount = .5(.5)^(4860/1620)
= .5(.5)^3 = .5^4 = .0625
.0625 g will be left
Well, if that hospital managed to keep that radium for almost 5000 years, I'm pretty sure it'll have entered the Guinness World Records for the "Longest-Lived Radium Sample in History"! But let's do the math to find out how much radium they'd have in the year 6836.
The half-life of radium is 1620 years, which means that after every 1620 years, half of the radium will decay. So, let's calculate:
Year 0: 0.5 grams
Year 1620: 0.25 grams (half decayed)
Year 3240: 0.125 grams (another half decayed)
Year 4860: 0.0625 grams (yet another half decayed)
Year 6480: 0.03125 grams (wow, this radium is persistent!)
Year 6836 🎉: Approximately 0.01514 grams (keep the party going!)
Now, remember, this calculation assumes that the hospital has somehow managed to keep the radium intact and accounted for during this entire time. Let's hope they've hired some radium-loving guardian angels to keep an eye on it!
To determine the amount of radium the hospital will have in the year 6836, we need to calculate the number of half-lives that have occurred between 1976 and 6836.
The half-life of radium is 1620 years, which means that after each half-life, half of the initial amount of radium remains. In this case, half of half a gram remains after one half-life, which is 0.5/2 = 0.25 grams.
Let's calculate the number of half-lives that have occurred between 1976 and 6836 by dividing the total number of years by the half-life:
6836 - 1976 = 4860 years
4860 / 1620 = 3 half-lives
Since three half-lives have occurred, the initial half gram of radium will be reduced by a factor of 2^3 = 8.
Therefore, the hospital will have 0.5 grams / 8 = 0.0625 grams of radium remaining in the year 6836.
To calculate how much radium the hospital will have in the year 6836, we need to determine the number of half-lives that have occurred since 1976 and use that to calculate the remaining quantity of radium.
The half-life of radium is 1620 years, which means that after every 1620 years, the quantity of radium will reduce by half.
To find the number of half-lives from 1976 to the year 6836, we subtract the starting year (1976) from the target year (6836) and divide it by the length of one half-life (1620 years):
Number of half-lives = (Target year - Starting year) / Half-life
Number of half-lives = (6836 - 1976) / 1620 = 3.5
Since we cannot have half a half-life, we round down to the nearest whole number, which in this case is 3.
Now that we know three half-lives have passed, we can calculate the remaining quantity of radium. Each half-life reduces the quantity by half, so we simply multiply the initial quantity by 1/2 three times:
Remaining quantity = Initial quantity * (1/2)^number of half-lives
Remaining quantity = 0.5 grams * (1/2)^3 = 0.5 grams * (1/8) = 0.0625 grams
Therefore, the hospital will have approximately 0.0625 grams of radium in the year 6836.