You are trying to determine the half-life of a new radioactive element you have isolated. You start with 1 gram, and 5 days later you determine that it has decayed down to 0.6 grams. What is its half-life? (Round your answer to three significant digits.)

Question

Mathematics

Casey Dominguez

12 October, 18:13

You are trying to determine the half-life of a new radioactive element you have isolated. You start with 1 gram, and 5 days later you determine that it has decayed down to 0.6 grams. What is its half-life? (Round your answer to three significant digits.)

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Brayden Boyd · Answer 1

The remaining mass mass of the substance at time t can be modeled with an exponential equation in this form:

m = ab^t,

where m is the remaining mass in grams, and t is the time in days.

We are given the mass at time zero and at time two.

Let t = 0.

10 = ab^0

a = 10

m = 10b^y

Let t = 2.

7 = 10b²

b² = 0.7

b ≈ 0.84

m ≈ 10 (0.84) ^t

Now to find the half-life, let m = 5 g, which is half of the initial mass. Solve for t.

5 = 10 (0.84) ^t

(0.84) ^t = 0.5

t ln (0.84) = ln (0.5)

t = ln (0.5) / ln (0.84) ≈ 3.9 days