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Ratling [72]
2 years ago
12

A substance with a half life is decaying exponentially. If there are initially 12grams of the substance and after 70 minutes the

re are 7 grams, after how many minutes will there be 2 grams remaining?
Mathematics
1 answer:
dybincka [34]2 years ago
4 0

Answer:

Step-by-step explanation:

Use the half life formula

N=N_0e^{kt} where N is the amount after decay, No is the initial amount, e is Euler's number, k is the decay constant, and t is the time in minutes. For us, the first equation looks like this:

7=12e^{k(70)} and we will solve that for k and then use that value of k in the second equation to find time.

Begin by dividing 7 by 12 to get

\frac{7}{12}=e^{k(70)} Now take the natural log of both sides since the natural log and that e are inverses. The e then disappears:

ln(\frac{7}{12})=k(70)

Plug the left side into your calculator and then set it equal to the right side, giving us:

-.5389965007=70k

Divide both sides by 70 to get the k value of

k = -.00769995

Now the second equation looks like this:

2=12e^{(-.00769995)t where t is our only unknown now that we know k.

Begin by dividing both sides by 12 to get

\frac{1}{6}=e^{-.00769995t and take the natural log of both sides again to eliminate the e:

ln(\frac{1}{6})=-.00769995t

Take care of the left side on your calculator to get

-1.791759469 = -.00769995t and divide both sides by -.00769995 to get

t = 232.697 minutes

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