You drink a beverage with 100 mg of caffeine. each hour, the caffeine in your system decreases by about 12%.

a. after 5 hours, how many caffeine left in your system.
b. how long until you have 50mg of caffeine?

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

answer: domain and range mean all possible values of x and y that could be substituted to the inequality. a solution means all possible values that make the inequality statement true. so while the range is still all real numbers, the solution is the shaded area, for which y will always be greater than the line of the function.

in simplified version   it would be x ≤-4

step-by-step explanation:

multiply each term in x ≤-4 by -1