Answer:
a. The amount of caffeine left is 52.77 mg
b. It will take about 5.42 hours
Step-by-step explanation:
* Lets solve it as an exponential decay
- Exponential decay: If a quantity decrease by a fixed percent at
regular intervals, the pattern can be depicted by this functions
y = a(1 - r)^x
# a = initial value (the amount before measuring growth or decay)
# r = growth or decay rate (most often represented as a percentage
and expressed as a decimal)
# x = number of time intervals that have passed
* Now lets solve the problem
∵ The initial amount of caffeine is 100 mg
∴ a = 100 mg
∵ The caffeine decreases by about 12% each hour
∴ r = 12/100 = 0.12
* Lets solve a.
a. ∵ x = 5 ⇒ the time interval
∵ The amount of caffeine left = a(1 - r)^x
∴ The amount of caffeine left = 100(1 - 0.12)^5
∴ The amount of caffeine left = 100(0.88)^5= 52.77 mg
* To find the time x use the linear logarithmic function
b. ∵ The amount of caffeine is 50 mg
∴ 50 = 100(1 - 0.12)^x ⇒ divide both sides by 100
∴ 50/100 = (0.88)^x
∴ 0.5 = (0.88)^x ⇒ take ln for each side
∴ ln(0.5) = ln(0.88)^x
∵ ln(a)^n = n ln(a)
∴ ln(0.5) = x ln(0.88) ⇒ divide both sides by ln(0.88)
∴ x = ln(0.5)/ln(0.88) = 5.4 years
* It will take about 5.42 hours
