The mass of particles of a certain radioactive chemical element (O Level Math Tuition Question)

Question from http://www.kiasuparents.com/kiasu/forum/viewtopic.php?f=29&t=8315&start=780

The mass of particles of a certain radioactive chemical element is halved every 10 months.  During a chemical experiment, the initial mass of particles of the chemical element is 3mg.

(i) write down an expression, in terms of t, for the mass of particles after t years.
(ii) Hence, find the value of t, if the mass is reduced to 0.046875 mg after t years.

Solution:

(i)

1\: \text{year} = 12\: \text{months}

Therefore, t\: \text{years}=12t\: \text{months}.

How many 10 months are there in t\: \text{years}? (Ans: \frac{12t}{10}=1.2t)

Hence, the mass of particles after t years is 3\times(0.5)^{1.2t} mg.

(ii)
We need to solve 3(0.5)^{1.2t}=0.046875.

Dividing by 3, we have (0.5)^{1.2t}=0.015625.

Ln both sides, we have 1.2t\ln{0.5}=\ln{0.015625}.

Hence, t=\frac{\ln{0.015625}}{1.2\ln{0.5}}=5.

 

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