Question

The model for radioactive decay is y=y0e−kt. A radioactive substance has a half-life of 200 years. If 45 grams are present today, in how many years will 25 grams be present? While solving this problem, round the value of k to seven decimal places. Round your answer to two decimal places.

Answer 1

Answer:

Therefore 25 grams will be present in 135.62 years.

Step-by-step explanation:

The model for radioactive decay is

$y=y_0e^{-kt}$

y=remaining amount of radioactive element.

$y_0$ = initial amount of radioactive element

Given that, the half life of the radioactive element is 200 years.

For half life, $y=\frac{1}{2}y_0$, t=200 years

$y=y_0e^{-kt}$

$\Rightarrow \frac12 y_0=y_0e^{-k.200}$

$\Rightarrow \frac12 =e^{-k.200}$

Taking ln both sides

$\Rightarrow ln|\frac12 |=ln|e^{-k.200}|$

$\Rightarrow ln|\frac12 |={-k.200}$

$\Rightarrow -200k=ln|\frac12|$

$\Rightarrow k=\frac{ln|\frac12|}{-200}$

⇒ k= 0.0034657

Given that,

y= 25 gram, y₀= 45 grams

$25=40e^{-0.0034657t}$

$\Rightarrow \frac{25}{40}=e^{-0.0034657t}$

Taking ln both sides

$\Rightarrow ln|\frac{25}{40}|=ln|e^{-0.0034657t}|$

$\Rightarrow ln|\frac{25}{40}|={-0.0034657t}$

$\Rightarrow t=\frac{ln|\frac{25}{40}|}{-0.0034657}$

⇒ t= 135.62 years.

Therefore 25 grams will be present in 135.62 years.