Question

radioisotopes

Answer 1

\[\huge (.875)^{\frac{x}{5.2}}=.5\] solve for \(x\)

Answer 2

From my website here's a formula:
half-life = (elapsed time * log 2) / log (begng amt/ ending amt)
half-life = (5.2 years * 0.30102999566) / log (100 / 12.5)
half-life = (5.2 years * 0.30102999566) / log (8)
half-life = (5.2 years * 0.30102999566) / 0.90308998699
half-life = 1.7333333333 years

OR a simpler way is
after 1 half life amt remaining is 50%
2 half lives = 25%
3 half lives = 12.5%

After 3 half lives = 5.2 years so we take the cube root of 5.2

1.732 years

http://www.1728.org/halflife.htm

Answer 3

@wolf1728 i had what you had the first time1.7 yr and the teacher says its 28 yr. So i don't know.
@satellite73 how did you get the .875?

Answer 4

i subtracted \(12.5\%\) from \(100\%\) lets solve it

Answer 5

oh okay thank you

Answer 6

that's where i was getting confused. so i get 28.2 yrs as the answer

Answer 7

\[(.875)^{\frac{x}{5.2}}=.5\\
\frac{x}{5.2}=\frac{\ln(.5)}{\ln(.875})\\
x=5.2\times \frac{\ln(.5)}{\ln(8.75)}\]

Answer 8

ok clearly i did something wrong

Answer 9

okay so it should be 1.7 yrs correct?

Answer 10

the solution is 27 years
i must have solved incorrectly let me see if i can find my mistake

Answer 11

oh no there is no mistake, i made a typo!!

Answer 12

\[(.875)^{\frac{x}{5.2}}=.5\\
\frac{x}{5.2}=\frac{\ln(.5)}{\ln(.875})\\
x=5.2\times \frac{\ln(.5)}{\ln(.875)}\]

Answer 13

i stupidly put 8.75 in the calculation instead of 8.75

Answer 14

http://www.wolframalpha.com/input/?i=5.2log%28.5%29%2Flog%28.875%29

Answer 15

or directly without solving
http://www.wolframalpha.com/input/?i=%28.875%29^%28x%2F%285.2%29%29%3D.5

Answer 16

i am not sure why your teacher got 28 years, i am sticking with 27

Answer 17

thank you both

Answer 18

It seems the original problem has been erased. I believe it was:

If 12.5% of a certain radioisotope decays in 5.2 years, what is the half life?

Okay, I see where we all went wrong. 12.5% DECAYS in 5.2 years so that means after 5.2 years, 87.5% of the original still REMAINS.

So using the formula:
half-life = (elapsed time * log 2) / log (begng amt/ ending amt)
half-life = (5.2 years * log 2) / log(100 / 87.5)
half-life = (5.2 * 0.30102999566) / log (1.1428571429)
half-life = (1.5653559774 / 0.057991947)
half life = 26.992643962 years

So, it seems that 27 years is the answer.