The answer is 5/8 milliliter are left in the bottle at lunch
Answer:
The initial mass of the sample was 16 mg.
The mass after 5 weeks will be about 0.0372 mg.
Step-by-step explanation:
We can write an exponential function to model the situation.
Let the initial amount be A. The standard exponential function is given by:
![P(t)=A(r)^t](https://tex.z-dn.net/?f=P%28t%29%3DA%28r%29%5Et)
Where r is the rate of growth/decay.
Since the half-life of Palladium-100 is four days, r = 1/2. We will also substitute t/4 for t to to represent one cycle every four days. Therefore:
![\displaystyle P(t)=A\Big(\frac{1}{2}\Big)^{t/4}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20P%28t%29%3DA%5CBig%28%5Cfrac%7B1%7D%7B2%7D%5CBig%29%5E%7Bt%2F4%7D)
After 12 days, a sample of Palladium-100 has been reduced to a mass of two milligrams.
Therefore, when x = 12, P(x) = 2. By substitution:
![\displaystyle 2=A\Big(\frac{1}{2}\Big)^{12/4}](https://tex.z-dn.net/?f=%5Cdisplaystyle%202%3DA%5CBig%28%5Cfrac%7B1%7D%7B2%7D%5CBig%29%5E%7B12%2F4%7D)
Solve for A. Simplify:
![\displaystyle 2=A\Big(\frac{1}{2}\Big)^3](https://tex.z-dn.net/?f=%5Cdisplaystyle%202%3DA%5CBig%28%5Cfrac%7B1%7D%7B2%7D%5CBig%29%5E3)
Simplify:
![\displaystyle 2=A\Big(\frac{1}{8}\Big)](https://tex.z-dn.net/?f=%5Cdisplaystyle%202%3DA%5CBig%28%5Cfrac%7B1%7D%7B8%7D%5CBig%29)
Thus, the initial mass of the sample was:
![A=16\text{ mg}](https://tex.z-dn.net/?f=A%3D16%5Ctext%7B%20mg%7D)
5 weeks is equivalent to 35 days. Therefore, we can find P(35):
![\displaystyle P(35)=16\Big(\frac{1}{2}\Big)^{35/4}\approx0.0372\text{ mg}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20P%2835%29%3D16%5CBig%28%5Cfrac%7B1%7D%7B2%7D%5CBig%29%5E%7B35%2F4%7D%5Capprox0.0372%5Ctext%7B%20mg%7D)
About 0.0372 mg will be left of the original 16 mg sample after 5 weeks.
Since 20 percent of 175 is 35, the total cost of the meal is 210. Dividing that between 5 people, each person would pay $42.
7/12 is the greatest value. Hope I helped you out :)