Question

An unknown radioactive element decays into non-radioactive substances. In 440 days the radioactivity of a sample decreases by 74 percent.
(a) What is the half-life of the element?
half-life: (days)

(b) How long will it take for a sample of 100 mg to decay to 91 mg?
time needed: (days)

Answer 1

Answer:

a) The half life of the element is 231 days.

b) It is going to take around 31.5 days for a sample of 100 mg to decay to 91 mg.

Step-by-step explanation:

The radioactivity of the sample can be modeled by the following exponential equation:

$R(t) = R(0)e^{-rt}$

In which t is the time in days, r is the decay rate and $R(0)$ is the initial radioactive percentage.

We have that:

In 440 days the radioactivity of a sample decreases by 74 percent.

This means that $R(440) = 0.26R(0)$.

This helps us find r.

$R(t) = R(0)e^{-rt}$

$0.26R(0) = R(0)e^{-440r}$

$e^{-440r} = 0.26$

Applying ln to both sides.

$\ln{e^{-440r}} = \ln{0.26}$

$-440r = -1.347$

$r = 0.003$

(a) What is the half-life of the element?

This is t when $R(t) = 0.50R(0)$

$R(t) = R(0)e^{-rt}$

$0.50R(0) = R(0)e^{-0.003t}$

$e^{-0.003t} = 0.5$

Again, we apply ln to both sides of the equality.

$\ln{e^{-0.003t}} = \ln{0.5}$

$-0.003t = -0.693$

$t = 231$

The half life of the element is 231 days.

(b) How long will it take for a sample of 100 mg to decay to 91 mg?

This is t when $R(t) = 0.91R(0)$

$R(t) = R(0)e^{-rt}$

$0.91R(0) = R(0)e^{-0.003t}$

$e^{-0.003t} = 0.91$

$\ln{e^{-0.003t}} = \ln{0.91}$

$-0.003t = -0.09$

$t = 31.44$

It is going to take around 31.5 days for a sample of 100 mg to decay to 91 mg.