Question

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??

Answer 1

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