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Tiffany is monitoring the decay of two radioactive compounds in test tubes at her lab. Compound A is continuously decaying at a rate of [tex]\(12\%\)[/tex] and compound B is continuously decaying at a rate of [tex]\(18\%\)[/tex]. Tiffany started with 30 grams of compound A and 40 grams of compound B.

Create a system of inequalities that can be used to determine when both compounds will be less than or equal to the same mass, [tex]\(M\)[/tex], where [tex]\(t\)[/tex] is time in weeks, [tex]\(P_A\)[/tex] is the initial amount of compound A, [tex]\(P_B\)[/tex] is the initial amount of compound B, and [tex]\(r\)[/tex] is the rate of decay.

Enter the inequalities in the field by replacing the values of [tex]\(P_A\)[/tex], [tex]\(P_B\)[/tex], and [tex]\(r\)[/tex]:

[tex]\[30 e^{-0.12t} \leq M\][/tex]

[tex]\[40 e^{-0.18t} \leq M\][/tex]



Answer :

GinnyAnswer
To determine when both compounds will be less than or equal to the same mass [tex]\( M \)[/tex] over time [tex]\( t \)[/tex] in weeks, we'll write a system of inequalities based on their continuous decay rates.

Given:
- Initial amount of compound [tex]\( A \)[/tex] ([tex]\( P_A \)[/tex]): 30 grams
- Continuous decay rate of compound [tex]\( A \)[/tex] ([tex]\( r_A \)[/tex]): [tex]\( -0.12 \)[/tex] per week (since 12% is the rate and decay implies a negative rate)
- Initial amount of compound [tex]\( B \)[/tex] ([tex]\( P_B \)[/tex]): 40 grams
- Continuous decay rate of compound [tex]\( B \)[/tex] ([tex]\( r_B \)[/tex]): [tex]\( -0.18 \)[/tex] per week (since 18% is the rate and decay implies a negative rate)

The general formula for the remaining amount of a compound undergoing continuous decay is:
[tex]\[ P e^{r t} \leq M \][/tex]

For compound [tex]\( A \)[/tex]:
[tex]\[ 30 e^{-0.12 t} \leq M \][/tex]

For compound [tex]\( B \)[/tex]:
[tex]\[ 40 e^{-0.18 t} \leq M \][/tex]

Therefore, the system of inequalities that can be used to determine when both compounds will be less than or equal to the same mass [tex]\( M \)[/tex] is:

[tex]\[ 30 e^{-0.12 t} \leq M \][/tex]
[tex]\[ 40 e^{-0.18 t} \leq M \][/tex]

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