Radioactive iodine treatment is so successful at treating hyperthyroidism that it has virtually replaced thyroid surgery. To the nearest full day, determine how long it will take for 400 millicuries of I-131, which has a half-life of 8 days, to decay to 3.125 millicuries.
56 days,
7 days,
128 days,
or
8 days
which one is it?
3.125/400 = (1/2)^(t/8)
(if t is in days)
7.8125*10^-3 = (1/2)^(t/8)
t/8 = Log(7.8125*19^-3)/Log(1/2)
= 7
t = 56 days
thank you so much!!
To determine the time it takes for the radioactive iodine to decay from 400 millicuries to 3.125 millicuries, we can use the formula for radioactive decay:
Amount remaining = Initial amount * (1/2)^(time/half-life)
Let's solve this step-by-step:
1. Initial amount = 400 millicuries
Amount remaining = 3.125 millicuries
Half-life = 8 days
2. Rearranging the formula, we have:
(1/2)^(time/half-life) = Amount remaining / Initial amount
3. Substituting the given values:
(1/2)^(time/8) = 3.125 / 400
4. Taking the logarithm (base 1/2) of both sides:
log(base 1/2)[(1/2)^(time/8)] = log(base 1/2)[3.125 / 400]
5. Simplifying the left side:
time/8 = log(base 1/2)(3.125 / 400)
6. Applying the change of base formula:
time/8 = log(3.125 / 400) / log(1/2)
7. Evaluating the logarithms using a calculator:
time/8 ≈ -1.1705 / -0.3010
8. Solving for time:
time ≈ 8 * (-1.1705 / -0.3010)
Calculating the approximation, we find:
time ≈ 31.4901 days
Rounding the result to the nearest full day, we have:
time ≈ 31 days
Therefore, it will take approximately 31 days (nearest full day) for 400 millicuries of I-131 to decay to 3.125 millicuries.
None of the given answer choices match the correct result.
To determine the time it will take for 400 millicuries of I-131 to decay to 3.125 millicuries, we need to use the concept of half-life.
The half-life of I-131 is given as 8 days, which means that in 8 days, half of the initial amount will have decayed. After another 8 days, half of what is remaining will decay, and so on.
To solve this problem, we can use the formula for exponential decay:
N(t) = N0 * (1/2)^(t/h),
where N(t) is the amount at time t, N0 is the initial amount, t is the time, and h is the half-life.
In our case, N0 is 400 millicuries, N(t) is 3.125 millicuries, and h is 8 days. We need to solve for t.
3.125 = 400 * (1/2)^(t/8)
Dividing both sides by 400, we get:
0.0078125 = (1/2)^(t/8)
Now, let's take the logarithm of both sides to get rid of the exponent:
log(0.0078125) = log((1/2)^(t/8))
Using logarithm properties, we can bring down the exponent:
log(0.0078125) = (t/8) * log(1/2)
Rearranging the equation to solve for t:
t/8 = log(0.0078125) / log(1/2)
Using a calculator, evaluate the right side:
t/8 ≈ -6.997
To solve for t, multiply both sides by 8:
t ≈ -6.997 * 8
t ≈ -55.98
The time cannot be negative, so we take the absolute value:
t ≈ 55.98
Therefore, to the nearest full day, it will take approximately 56 days for 400 millicuries of I-131 to decay to 3.125 millicuries.
Therefore, the correct answer is 56 days.