Answer:
38.3 mg
Step-by-step explanation:
Let's denote the initial amount of caffeine in the coffee as [tex]\sf C_0 [/tex].
The formula for the remaining amount of caffeine after [tex]\sf t [/tex] hours, given a decay rate of 15% per hour, is:
[tex]\sf C(t) = C_0 \times (1 - r)^t [/tex]
where [tex]\sf r [/tex] is the decay rate (as a decimal), and [tex]\sf t [/tex] is the time in hours.
In this case, [tex]\sf r = 0.15 [/tex] (15% as a decimal), and [tex]\sf t = 4 [/tex] hours.
The remaining amount of caffeine after 4 hours is given as 20 mg:
Now,
Substitute the given value:
[tex]\sf 20 = C_0 \times (1 - 0.15)^4 [/tex]
Now, solve for [tex]\sf C_0 [/tex]:
[tex]\sf 20 = C_0 \times (0.85)^4 [/tex]
[tex]\sf 20 = C_0 \times 0.52200625 [/tex]
[tex]\sf C_0 = \dfrac{20}{0.52200625} [/tex]
[tex]\sf C_0 \approx 38.31371751 [/tex]
[tex]\sf C_0 \approx 38.3 \textsf{ mg (in nearest tenth)}[/tex]
Therefore, the initial amount of caffeine in Alex's cup of coffee was approximately 38.3 mg.
