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# half life

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An 80-gram sample of a radioactive substance decayed to 10 grams after 300 minutes. find the half-life of the substance

May 23, 2017

#1
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10 = 80 x 2^-(300/h), solve for h

Solve for h over the real numbers:
10 = 5 2^(4 - 300/h)

10 = 5 2^(4 - 300/h) is equivalent to 5 2^(4 - 300/h) = 10:
5 2^(4 - 300/h) = 10

Divide both sides by 5:
2^(4 - 300/h) = 2

Take reciporicals of both sides:
2^(300/h - 4) = 1/2

Take the logarithm base 2 of both sides:
300/h - 4 = -1

Bring 300/h - 4 together using the common denominator h:
-(4 (h - 75))/h = -1

Multiply both sides by h:
-4 (h - 75) = -h

Expand out terms of the left hand side:
300 - 4 h = -h

Subtract 300 - h from both sides:
-3 h = -300

Divide both sides by -3:
Answer: | h = 100 minutes - the half-life.

May 23, 2017
#2
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An 80-gram sample of a radioactive substance decayed to 10 grams after 300 minutes. find the half-life of the substance.

10 = 80*2^-(300/h), solve for h. Divide both sides by 80,

0.125 = 2^-(300/h)   take the ln of both sides,

-2.0794415... =-(300/h)*0.6931472..... divide both sides by 0.6931472.....

-3 = -(300/h)  cross multiply

-3h = -300 divide both sides by -3

h =-300/-3

h= 100 minutes - the half-life of the radioactive substance.

May 23, 2017