Suppose you have a cache of radium, which has a half-life of approximately 1590 years. How long would you have to wait for 1/5 of it to disappear?
Oh, radium, the glowing gem of half-lives! Well, if radium has a half-life of approximately 1590 years, then it takes quite a while for things to start disappearing. If you want to know when 1/5 of your radium will vanish into thin air, let me calculate it for you... (beep boop beep)
Here's the mathematically lighthearted answer: Since radium has a half-life of 1590 years, we can divide this number by 0.693, which represents the natural logarithm of 2. This gives us the decay constant of approximately 2290 years.
So, to find out when 1/5 of your radium will bid you farewell, multiply the half-life by the logarithm of 5 (since 1/5 is 5 times smaller than the original quantity). Voilà! You'll have your answer.
Meanwhile, while we're waiting, let's hope the radium doesn't decide to put on a magic trick show and pull a disappearing act sooner than expected!
To determine the time it takes for 1/5 of the radium to disappear, we can use the concept of exponential decay and the formula for the half-life of a substance.
1. Start by calculating the number of half-lives needed for 1/5 (or 20%) of the radium to decay. Since each half-life reduces the amount by half, we need to find the number of half-lives it takes for 1/5 to become 1/2.
1/5 = (1/2)^(n), where 'n' is the number of half-lives.
2. Take the logarithm of both sides of the equation to solve for 'n':
log(1/5) = log[(1/2)^n]
log(1/5) = n * log(1/2)
3. Divide both sides of the equation by log(1/2) to solve for 'n':
n = log(1/5) / log(1/2)
4. Use a calculator to evaluate the right side of the equation:
n ≈ 2.3219
5. Since each half-life corresponds to approximately 1590 years, multiply 'n' by the half-life:
time = n * half-life
time ≈ 2.3219 * 1590
time ≈ 3691.62 years
Therefore, you would have to wait for approximately 3691.62 years for 1/5 of the radium to disappear.
To determine how long it would take for 1/5 of the radium to disappear, we can use the concept of half-life. The half-life is the time it takes for half of a radioactive substance to decay.
Given that the half-life of radium is approximately 1590 years, we can calculate the number of half-lives it would take for 1/5 of the radium to decay.
Since we want to find 1/5 of the radium, this means we need to determine how many half-lives it takes for the remaining 4/5 of the radium to decay.
Let's calculate the number of half-lives:
1. Start with the fraction of radium remaining: 4/5.
2. Divide the fraction by 1/2 (representing one half-life): (4/5) ÷ (1/2) = 8/5.
3. Calculate the logarithm base 2 of the resulting fraction: log2(8/5) ≈ 0.678.
The number 0.678 represents the number of half-lives required for 4/5 of the radium to decay. Each half-life is 1590 years, so now we can find the time it would take for 1/5 of the radium to disappear:
Time = (Number of half-lives) * (Half-life duration)
≈ 0.678 * 1590 years.
Calculating this:
Time ≈ 1078 years.
Therefore, you would have to wait approximately 1078 years for 1/5 of the radium to disappear.
just solve
2^(-t/1590) = 1/5
since 2^-2 = 1/4, you should expect an answer something more than two half-lives.