Cobalt-60 is a strong gamma emitter that has a half-life of 5.26 yr. The cobalt-60 in a radiotherapy unit must be replaced when its radioactivity falls to 75% of the original sample. If an original sample was purchased in October 2005, when will it be necessary to replace the cobalt-60?
i have no idea how to start this question please help!
after t years, the fraction left is
2^(-t/5.26)
so, just solve
.75 = 2^(-t/5.26)
t = 2.183 years
2.183 years = 2 yrs + 2.2 months
so, it gets tossed in Dec 2007, or Jan 2008 (if it was purchased after Oct. 25)
Well, this is quite a radioactive situation we have here! Don't worry, I'm here to enlighten you with a radioactive joke while we tackle this problem.
Why don't scientists trust atoms?
Because they make up everything!
Now, let's get back to business. To solve this problem, we need to determine how many half-lives have passed until the radioactivity falls to 75% of the original sample.
Since cobalt-60 has a half-life of 5.26 years, we can calculate the number of half-lives that have passed by dividing the total time since October 2005 by the half-life of cobalt-60.
So, let's do the math. October 2005 to the present day would be 2021 – 2005 = 16 years. Since each half-life is 5.26 years, the number of half-lives that have passed is 16 / 5.26 ≈ 3.04.
Now that we know that approximately 3.04 half-lives have passed, we need to calculate the remaining radioactivity. Since each half-life reduces the radioactivity by half, the remaining radioactivity is 100% / 2^3.04 ≈ 44.03% of the original sample.
Since we need to replace the cobalt-60 when the radioactivity falls to 75% of the original sample, we know we have to replace it before it reaches 44.03%. Therefore, it's necessary to replace the cobalt-60 sometime between October 2005 and now.
Hope that helps! If you need any further assistance, feel free to ask.
To solve this problem, we can use the concept of exponential decay.
Given:
Half-life of cobalt-60 = 5.26 years
Radioactivity must fall to 75% of the original sample
Step 1: Calculate the number of half-lives that have occurred since October 2005.
To do this, divide the elapsed time by the half-life:
Elapsed time = Current year - Year of purchase
Step 2: Calculate the remaining radioactivity after this amount of time.
To do this, use the formula:
remaining radioactivity = original radioactivity × (1/2)^(number of half-lives)
Step 3: Determine when the radioactivity falls to 75% of the original sample.
Set up an equation:
remaining radioactivity = 75% of the original radioactivity
Step 4: Solve the equation to find the time needed to replace the cobalt-60.
Let's calculate the approximate date of replacement:
Step 1: Calculate the number of half-lives
Elapsed time = 2021 - 2005 = 16 years
Number of half-lives = elapsed time / half-life = 16 years / 5.26 years per half-life ≈ 3.04 half-lives
Step 2: Calculate the remaining radioactivity after this amount of time
remaining radioactivity = 1 × (1/2)^(3.04 half-lives) ≈ 0.419
Step 3: Determine when the radioactivity falls to 75% of the original sample
remaining radioactivity = 75% of original radioactivity
0.419 = 75% × 1
0.419 = 0.75
Step 4: Solve the equation to find the time needed to replace the cobalt-60
Since the radioactivity is already less than 75%, it is necessary to replace the cobalt-60 at the current time.
Therefore, the cobalt-60 needs to be replaced in the current year.
To determine when the cobalt-60 needs to be replaced, we need to calculate the time it takes for its radioactivity to decrease to 75% of the original sample. We can use the concept of half-life to solve this problem.
1. Start by determining the number of half-lives required for the radioactivity to fall to 75% of its original value. Since the half-life of cobalt-60 is 5.26 years, we divide the total time by the half-life.
Total time = Year of replacement - Year of purchase
Total time = Year of replacement - 2005
2. Once we have the total time in years, we calculate the number of half-lives needed. Since the activity falls to 75% of the original, we can calculate the fraction remaining after each half-life using the formula: Remaining fraction = (0.5)^(number of half-lives).
Remaining fraction = 75% = 0.75
Remaining fraction = (0.5)^(number of half-lives)
3. Solve for the number of half-lives:
(0.5)^(number of half-lives) = 0.75
Take the logarithm of both sides to solve for the number of half-lives:
log base 0.5 (0.75) = number of half-lives
4. Now we can calculate the number of years required for the activity to fall to 75%:
Number of years = number of half-lives * half-life
Number of years = (number of half-lives) * 5.26
Substitute the value of the number of half-lives calculated in step 3 into this equation.
5. Finally, add the calculated number of years to the year of purchase (2005) to find out when the replacement is necessary.
Now you can plug in the values and calculate the answer following these steps.