Mathematics: The element aopsium has a half-life of 120 years. after a certain sample of aopsium was stored for 600 years, only 7 grams was left. how many grams were in the original sample?

A, C, D, on Edge 2020Step-by-step explanation:...

The element aopsium has a half-life of 120 years.

lakaralake11

31.12.2021

Initial amount of the element taken was 224 grams.

Step-by-step explanation:

Half life of the element AoPSium is 120 years.

We know the radioactive decay is represented by the formula

$A_{t}=A_{0}(e)^{-kt}$

where A(t) = Element remaining after time t

A(0) = Initial amount

t = time in years

k = decay constant

$\frac{A_{0} }{2}=A_{0}(e)^{-k\times 120}$

$\frac{1}{2}=e^{-120k}$

Now we take natural log on both the sides of the equation

$ln(\frac{1}{2})=ln[(e)^{-120k}]$

-0.69314 = - 120k

k = $\frac{0.69314}{120}$

k = 0.005776

Now we have to calculate the amount taken of the element after 600 years when amount remaining was 7 grams.

$A_{t}=A_{0}(e)^{-kt}$

7 = $A_{0}(e)^{-(0.0057762\times 600)}$

7 = $A_{0}(e)^{-3.4657}$

$A_{0}=7\times e^{(3.4657)}$

$A_{0}=31.99885\times 7$

$A_{0}=224$ grams

Therefore, Initial amount of the element taken was 224 grams.

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