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# Pre Calc

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A 20 gram sample of the radioactive element mathtestium takes 10 years to decrease to 15 grams. Find the half life of mathtestium. (You may use e, ln, or log to express your answer)

THIS IS A TEST QUESTION, can someone please solve this as if you were NOT allowed to use a calculator????

thank you

Dec 16, 2019

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15   =  20 e^(k* 10)    divide both sides by   20

3/4  = e^(10k)       taket the Ln of both sides

Ln (3/4) = Ln e^(10 k)    and we can write

Ln (.75)  = (10k) * Ln e        [ Ln e  =1....so....we can ignore this  ]

Ln (.75) /=  10k     divide both sides by 10

Ln (.75)/10  =  k

And we want to know when 10  grams remain....so....

10  = 20e^(t * Ln(.75) / 10 )           divide both sides by 20

.5  = e^(t * Ln (.75)/10 )          using a similar proceedure  as above

Ln .5  =  t * Ln (.75) /  10         multiply both sides by  10 / Ln (.75) and we get that

10 * Ln (.5) /  Ln (.75)  =  t

This is the exact  answer........you will need a calculator from here   [ the answer is ≈  24.09 yrs  ]   Dec 16, 2019