Iodine-131 is a radioactive isotope used in the treatment of thyroid conditions. It has a half-life of 8 days. Half-life is the amount of time it takes for half of the substance to decay (or disappear). If a patient is given 20 mg of iodine-131, how much of the substance will remain in the body after 32 days?
How do I set this up to solve? y = 20( )^x
How do I represent half-life and then put it into my calculator to solve??
N(t) = N₀ e ⁻ᵏᵗ
given:
N₀ = 20
th = 8
t = 32
N(t) is the amount after the time t
N₀ is the initial amount
th is the half-life
After half-life there will be twice less the initial quantity:
N(th) = N₀ / 2 = N₀ e ⁻ ᵏ ᵗʰ
First, find the constant k
N₀ / 2 = N₀ e ⁻ ᵏ ᵗʰ
Divide both sides by N₀
1 / 2 = e ⁻ ᵏ ᵗʰ
Take the ln of both sides
ln ( 1 / 2 ) = - k th
Divide both sides by - th
- ln ( 1 / 2 ) / th = k
k = - ln ( 1 / 2 ) / th
Plugging this into the initial equation, we obtain that:
N(t) = N₀ e ⁻ᵏᵗ = N₀ e^ - [ - ln ( 1 / 2 ) ∙ t / th ]
N(t) = N₀ e^ [ ( ln ( 1 / 2 ) ∙ t / th ]
Since:
e^ ln ( 1 / 2 ) = 1 / 2
N(t) = N₀ ∙ ( 1 / 2 ) ^ t / th
Plug in the given values and find the unknown one.
N(t) = N₀ ∙ ( 1 / 2 ) ^ t / th
N(t) = 20 ∙ ( 1 / 2 ) ^ 32 / 8
N(t) = 20 ∙ ( 1 / 2 ) ^ 4
N(t) = 20 ∙ 1 / 16 = 20 / 16 = 4 ∙ 5 / 4 ∙ 4 = 5 / 4 = 1.25 mg
To set up the equation and represent the half-life, you can use the formula:
y = initial amount * (1/2)^(x/h)
Where:
- y is the remaining amount of substance after a certain time (in this case, after 32 days)
- initial amount is the starting amount of the substance (20 mg)
- (1/2) is the fraction represents decay during one half-life
- x is the number of time intervals that have passed (in this case, the number of half-lives)
- h is the half-life of the substance (8 days)
Substituting the given values into the equation:
y = 20 * (1/2)^(32/8)
Simplifying further:
y = 20 * (1/2)^4
Now, you can enter this equation in your calculator to solve for the remaining amount of the substance.
To set up the equation to solve for the amount of iodine-131 remaining in the body after 32 days, you can use the formula:
y = 20 * (1/2)^(x/t)
Where:
- y represents the remaining amount of iodine-131 after a given period of time
- 20 mg is the initial amount of iodine-131 given to the patient
- x is the number of days that have passed
- t is the half-life of iodine-131, which is 8 days
By substituting the values into the equation, it becomes:
y = 20 * (1/2)^(x/8)
To calculate the remaining amount using a calculator, here's what you can do:
1. Enter "1/2" into your calculator, either as "0.5" or using the fraction key, depending on the type of calculator you have.
2. Raise this value to the power of (32/8) to represent the 32 days.
3. Multiply the result by 20.
For example, on a regular scientific calculator, you can follow these steps:
1/2 = 0.5
(32/8) = 4
0.5^4 = 0.0625
0.0625 * 20 = 1.25 mg
So, after 32 days, approximately 1.25 mg of iodine-131 will remain in the patient's body.