Question

Jordan works in a science lab where he is studying the behavior of a certain unstable isotope. He has 240 milligrams of the sample, and the amount of the substance remaining in the sample decreases at a rate of 8% each day. After t days, there are less than 115 milligrams of the substance remaining. Which inequality represents this situation, and after how many days will the amount of the sample be less than 115 milligrams?

A. 115(0.92)t < 240; 10 days
B. 240(1.08)t < 115; 9 days
C. 240(0.92)t < 115; 9 days
D. 115(1.08)t < 240; 8 days

Answer 1

Let the initial mass be represented by
`m_0`. Therefore,
`m_0=240 mg`.

Let the final mass (or present mass be represented by
`m`. Therefore,
`m=115mg`.

Now, we are given that the rate of decay is 8% or 0.008 per day. Therefore, the amount of substance remaining after each day will be
`100\%-8\%=92\%=0.92`. Let us represent this by
`r_d`.

Please note that the equation guiding such decays is always given by:

`m=m_0(r_d)^t`, where t is the time.

Thus, the above equation will give in our case:

`115=240(0.92)^t`

`(115)/(240)\approx0.4792=(0.92)^t`

Taking natural logarithm on both sides we get:

`ln(0.4792)=t* ln(0.92)`

Therefore,
`t=(ln(0.4792))/(ln(0.92))`

`t\approx8.823` days<9 days

Thus, in 9 days there will be further decay and the sample left will definitely be less than 115 milligrams.

The only option which matches this reality is Option C. Therefore, Option C is the correct answer.