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

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 [tex]$A$[/tex], [tex]$P_B$[/tex] is the initial amount of compound [tex]$B$[/tex], 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]\[30 e^{-0.12 t} \leq M\][/tex]
[tex]\[40 e^{-0.18 t} \leq M\][/tex]



Answer :

GinnyAnswer
Tiffany starts with 30 grams of compound [tex]$A$[/tex] and 40 grams of compound [tex]$B$[/tex].

The decay rates are:
- Compound [tex]$A$[/tex]: 12% per week
- Compound [tex]$B$[/tex]: 18% per week

This gives:
- [tex]\( P_A = 30 \)[/tex]
- [tex]\( P_B = 40 \)[/tex]
- [tex]\( r_A = -0.12 \)[/tex]
- [tex]\( r_B = -0.18 \)[/tex]

The general form of the inequalities is:
[tex]\[ P_A e^{r_A t} \leq M \][/tex]
[tex]\[ P_B e^{r_B t} \leq M \][/tex]

Plugging in the values, we get:
[tex]\[ 30 e^{-0.12 t} \leq M \][/tex]
[tex]\[ 40 e^{-0.18 t} \leq M \][/tex]

So, the system of inequalities is:
[tex]\[ 30 e^{-0.12 t} \leq M \][/tex]
[tex]\[ 40 e^{-0.18 t} \leq M \][/tex]

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