Question

Mr. Greene likes to drink a cup of green tea every morning. One cup has about 35 mg of caffeine in it, and the amount of caffeine left in the bloodstream decreases by 23% every hour.

a.) Write a function that models the situation.
b.) How much caffeine will be in his system after 4 hours?
c.) BONUS: Approximately how long will it take for there to be only 5 mg of caffeine left

Answer 1

The exponential decay function that models the situation is f(x) = 35 * (0.77)^x. After 4 hours, there will be approximately 15.50 mg of caffeine in Mr. Greene's system. It will take approximately 9.62 hours for there to be only 5 mg of caffeine left.

Step-by-step explanation:

a) The exponential decay function that models the situation is:

f(x) = 35 * (0.77)^x

where x is the number of hours elapsed.

b) To find the amount of caffeine in Mr. Greene's system after 4 hours, we need to evaluate the function f(4):

f(4) = 35 * (0.77)^4 ≈ 15.50 mg

So, there will be approximately 15.50 mg of caffeine in his system after 4 hours.

c) To find approximately how long it will take for there to be only 5 mg of caffeine left, we need to solve the equation:

35 * (0.77)^x = 5

First, divide both sides by 35:

(0.77)^x = 5/35 ≈ 0.1429

Then, take the natural logarithm of both sides:

x * ln(0.77) = ln(0.1429)

Finally, divide both sides by ln(0.77) and solve for x:

x ≈ ln(0.1429) / ln(0.77) ≈ 9.62 hours

So, it will take approximately 9.62 hours for there to be only 5 mg of caffeine left.

Answer 2

A. Y=k*x
K=23/100=0.23
Y=0.23*x (x is time by an hour)

B. X=4
Y= 0.23*4 = 0.92 mg

C. 5 = 0.23*x
5/0.23= 21.7 hour