Question

The function f(h)=m(1/2)^h gives the mass, m, of a radioactive substance remaining after h half-lives. Iron has a half-life of 2.7 years. Which equation gives the mass of a 200 mg Iron sample remaining after y years, and approximately how many milligrams remain after 12 years?

1: f(x) = 2.7(0.5)200; 1.7 mg
2: f(x) = 200(0.5)12; 30.8 mg
3: f(x) = 200(0.5)12; 30.8 mg
4: f(x) = 200(0.185)12; 3.2 mg
5: f(x) = 200(0.774)12; 9.2 mg

Answer 1

The answer is D

Explanation:

Answer 2

If a certain initial amount, A₀, of material decays with a half-life of h, the amount of material that remains at time t is given by the exponential decay model


`A(t) = A_(0) ( (1)/(2) )^{ (t)/(h) }`

==================================================
Part (a)
Which equation gives the mass of a 200 mg Iron sample remaining after y years?

∴ A₀ = 200 mg , t = y and h = 2.7 years
so, the equation will be:

∴
`f(y) =200( (1)/(2) )^{ (y)/(2.7) }`

====================================================

Part (b):
How many milligrams remain after 12 years?


By substitute with y = 12 in the equation obtained from (a)
∴
`f(12) =200( (1)/(2) )^{ (12)/(2.7) } \\ f(12) = 200( ((1)/(2))^{ (1)/(2.7) } )^(12) \\ f(12)= 200(0.774)^(12) \\ f(12) = 200 * 0.046 \\ f(12) = \framebox{9.2} \ mg`


So, the correct answer is option ⇒ 5: f(x) = 200(0.774)^12; 9.2 mg