What is the half life ?
Half-life is the time required for a quantity to reduce to half its initial value.
The term is used generally to characterize any type of exponential decay. When solving half-life problems, use the following formula:
A = Amount remaining
A_{0} = Initial amount
t = time elapsed
H = half life
Problem 1 :
The half-life of carbon-14 is known to be 5720 years. If 300 grams of carbon-14 are stored for 1200 years, how many grams will remain?
Solution :
Initial amount A_{0 }= 300 grams
Half life time (H) = 5720
Time elapsed (t) = 1200
Approximately 260 grams will remain.
Problem 2 :
When Sophia drinks a brewed cup of coffee, she ingests 130 mg of caffeine into her system.
The half-life of caffeine in a typical adult is 5.5 hours. How much caffeine will be in her system 4 hours after she drinks the cup of coffee?
Solution :
Initial amount A_{0 }= 130 mg
Half life time (H) = 4
Time elapsed (t) = 5.5
Problem 3 :
A colony of bacteria decays so that the population days from now is given by
n(t) = 1000(1/2)^{t/4}
a. What is the amount present when t = 0?
b. How much will be present in 4days?
c. What is the half-life?
Solution :
n(t) = 1000(1/2)^{t/4}
a) When t = 0
n(0) = 1000(1/2)^{0/4}
= 1000(1)
= 1000
Present amount of bacteria is 1000.
b. When t = 4
n(4) = 1000(1/2)^{4/4}
= 1000(1/2)^{1}
= 1000(0.5)
= 500
c) n(t) = 1000(1/2)^{t/4}
Initial amount = 1000, After half life time, it will become 500.
500 = 1000(1/2)^{t/4}
1/2 = (1/2)^{t/4}
(1/2)^{1} = (1/2)^{t/4}
t/4 = 1
t = 4
So, after 4 years the amount of bacteria will become half.
Problem 4 :
The half-life of a radioactive isotope is 4 days. If 3.2 kg are present now, how much will be present after:
a. 4 days?
b. 8 days?
c. 20 days
d. t days?
Solution :
Initial amount A_{0 }= 3.2 kg
Half life time (H) = 4
Time elapsed (t) =
a. 4 days
b. 8 days
c. 20 days
d. t days
Problem 5 :
The half life of radium is about 1600 years. If 1 kg is present now, how much will be present after
a) 3200 years b) 16000 years c) 800 years
Solution :
Initial amount A_{0 }= 1 kg
Half life time (H) = 1600
Time elapsed (t) = in years
a) 3200 years
b) 16000 years
c) 800 years
Problem 6 :
When a certain medicine enters the bloodstream, it gradually dilutes, decreasing exponentially with a half-life of 3 days. The initial amount of the medicine in the bloodstream is A_{0} milliliters. What will the amount be 30 days later?
Solution :
Initial amount = A_{0}
Half life time (H) = 3
Time elapsed (t) = 30 days
Problem 7 :
A culture starts with 1500 bacteria and the number doubles every 30 minutes.
In the half-life formula, use a 2 instead of 1/2 to find a function that models the number of bacteria at time t
b. Find the number of bacteria after 2 hours.
c. After how many minutes will there be 4000 bacteria?
Solution :
In 2 hours, we have 4 times 30 minutes
b)
c)
After 42.50 minutes 4000 bacteria will be there.
Dec 08, 23 08:03 AM
Dec 08, 23 07:32 AM
Dec 08, 23 07:10 AM