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
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??
1 answer