Question

Nickel-63 is a radioactive substance with a half-life of about 100 years. An artifact had

9.8 milligrams of nickel-63 when it was first measured. Write an equation to represent
the mass of nickel-63, in milligrams, as a function of:
a. tt, time in years
b. dd, time in days

Answer 1

a) x = 9.8 × 0.5^(t/100)

b) x = 9.8 × 0.5^(d/36500)

Explanation:

a) x = 9.8 × 0.5^(t/100)

100 years = 365×100 days

= 36500 days

b) x = 9.8 × 0.5^(d/36500)

Answer 2

see below

Explanation:

The equation for half life is

n = no e ^ (-kt)

Where no is the initial amount of a substance , k is the constant of decay and t is the time

no = 9.8

1/2 of that amount is 4.9 so n = 4.9 and t = 100 years

4.9 = 9.8 e^ (-k 100)

Divide each side by 9.8

1/2 = e ^ -100k

Take the natural log of each side

ln(1/2) = ln(e^(-100k))

ln(1/2) = -100k

Divide each side by -100

-ln(.5)/100 = k

Our equation in years is

n = 9.8 e ^ (ln.5)/100 t)

Approximating ln(.5)/100 =-.006931472

n = 9.8 e^(-.006931472 t) when t is in years

Now changing to days

100 years = 100*365 days/year

36500 days

Substituting this in for t

4.9 = 9.8 e^ (-k 36500)

Take the natural log of each side

ln(1/2) = ln(e^(-36500k))

ln(1/2) = -36500k

Divide each side by -100

-ln(.5)/36500 = k

Our equation in years is

n = 9.8 e ^ (ln.5)/36500 d)

Approximating ln(.5)/365=-.00001899

n = 9.8 e^(-.00001899 d) when d is in days