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At the beginning of an experiment, a scientist has 176 grams of radioactive goo. After 135 minutes, her sample has decayed to 11 grams. What is the half-life of the goo in minutes? Find a formula for G ( t ) , the amount of goo remaining at time t . G ( t ) = How many grams of goo will remain after 32 minutes?

User Rishabh Deep Singh
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1 Answer

10 votes
10 votes

Answer:

  • half-life: 33.75 minutes
  • formula: G(t) = 176(1/2)^(t/33.75)
  • G(32) ≈ 91.2 g

Explanation:

Given 176 grams of goo decays to 11 grams in 135 minutes, you want the half-life, a formula for the remaining amount, and the amount remaining after 32 minutes.

Formula

The formula for the decay can be written as ...

remaining = (initial amount)×(decay factor)^(t/period)

where "period" is the period that results in a remaining amount of ...

(initial amount)(decay factor)

Here, the initial amount is given as 176 grams. The decay factor will be (11/176) after a period of 135 minutes. This means our equation can be written as ...

G(t) = 176(11/176)^(t/135)

G(t) = 176(1/16)^(t/135) . . . . simplify the decay factor

G(t) = 176((1/2)^4)^(t/135) . . . . recognizing 16 = 2^4

G(t) = 176(1/2)^(t/33.75) . . . . . in terms of half-life

Half-life

The half-life is the period associated with a decay factor of 1/2. Here, we have found it to be 33.75 minutes

The half-life is 33.75 minutes.

Remaining

After 32 minutes, the amount remaining is ...

G(32) = 176(1/2)^(32/33.75) ≈ 91.22 . . . . grams

About 91.2 grams of goo will remain after 32 minutes.

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At the beginning of an experiment, a scientist has 176 grams of radioactive goo. After-example-1
User Espezy
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