Question

1) Danny consumes an energy drink that contains caffeine. After consuming the energy drink, the amount of caffeine in Danny's body decreases exponentially. The 10-hour decay factor for the number of mg of caffeine in Danny's body is 0.2484.
a) What is the 5-hour growth/decay factor for the number of mg of caffeine in Danny's body?
b) what is the 1 hour factor?
c) If there were 157 mg of caffeine in Danny's body 1.22 hours after consuming the energy drink, how many mg of caffeine is in Danny's body 2.22 hours after consuming the energy drink?


2) The population of a town increased by 2.76% per year from the beginning of 2000 to the beginning of 2010. The town's population at the beginning of 2000 was 74,780.
a) 1 yr growth factor?
b) Define a function f to represent the town's population f(t) in terms of the number of years t since the beginning of 2000.
c) what is the population at the beginning of 2010?

Answer 1

1) ans

Given

10 hour decay factor = 0.2484

Let us calculate the 1 hour decay factor first

1 hour decay factor = (10 hour decay factor)^(1/10 )

= (0.2484)^(1/10 )

0.86999

a) 5 hour decay factor = (1 hour decay factor)^5

= (0.86999)^5

= 0.49839

b) 1 hour decay factor = 0.86999[ As calculated above]

c) Let the initial amount of caffeine be x mg

Amount of caffeine after 1.22 hours = Initial amount * (1 hour decay factor)^1.22 = x (0.86999)^1.22

Amount of caffeine after 2.22 hours = Initial amount * (1 hour decay factor)^2.22

= x (0.86999)^2.22

= x (0.86999)^(1.22 + 1)

= x[ (0.86999)^1.22* ](0.86999)

= Amount of caffeine after 1.22 hours * (0.86999)

= 157 * 0.86999 mg = 136.588 mg

2)ans