Thread: Help me with these 2 problems

1. It is known that Carbon-14 atoms decay in an exponential manner. It takes about 5730 years for Carbon-14 atoms to decay to half their original amount. How many years will it take for the Carbon-14 atoms inside an object to decay to one-eighth of their original amount?

2. In a colony of bacteria, the rate at which the number of bacteria grows increases linearly with time (i.e. the number of new bacteria produced per unit time increases proportionally with time). Initially at time t = 0 hour there are 5 million bacteria in the colony and the initial rate of bacteria produced is 1 million per hour. At time t = 5 hours, the growth rate of the bacteria has increased to 6 million bacteria per hour.
a. What is the growth rate of bacteria at t = 2 hours?
b. What is the total number of bacteria at time t = 5 hours?
c. What is the average number of bacteria produced every hour during this 5 hours interval?

Perhaps these qns may seem basic, but i dont know why i just couldnt get the answers. Pls help me.

Originally Posted by dorwei92

1. It is known that Carbon-14 atoms decay in an exponential manner. It takes about 5730 years for Carbon-14 atoms to decay to half their original amount. How many years will it take for the Carbon-14 atoms inside an object to decay to one-eighth of their original amount?

What is the meaning of "half-life"? If it takes 5,730 years for half to disappear, how many more years will it take for half of what is left to disappear (leaving you with one-fourth of the original amount)? Then how many more years will it take to get down to one-eighth of the original amount?

Originally Posted by dorwei92

2. In a colony of bacteria, the rate at which the number of bacteria grows increases linearly with time (i.e. the number of new bacteria produced per unit time increases proportionally with time). Initially at time t = 0 hour there are 5 million bacteria in the colony and the initial rate of bacteria produced is 1 million per hour. At time t = 5 hours, the growth rate of the bacteria has increased to 6 million bacteria per hour.
a. What is the growth rate of bacteria at t = 2 hours?
b. What is the total number of bacteria at time t = 5 hours?
c. What is the average number of bacteria produced every hour during this 5 hours interval?

If the growth rate is linearly increasing, then you can find a linear formula (that is, a straight-line equation) for the change in the growth rate.

You are given the data points (t, r) = (0, 1) and (t, r) = (5, 6), where "r" has the units "millions per hour". What is the slope of the line through these points? What then is the equation of that line?

Once you have the line equation, plug "2" in for "t" to find the answer to part (a).

For part (b), I don't know if you have learned anything more complex than shown above. If not, just do the computations by hand: At t = 0, you have 5 millions (m) and know that 1 m will be added. This means that t = 1 starts with 5 + 1 = 6 m, and you know (from your equation) how many will be added. Continue until you have found the required total value.

For part (c), you can find the values of r for each value of t, and then do the "sum and divide" thing to find the average value.

If you get stuck, please reply showing your work and reasoning so far. Thank you!

For the first question, i still dont really understand.
My answer was 17190 but the worksheet answer was 17109.
My working:
Let the original amount be 1000,
1/2*1000 = 500 (take 5730 years to decay)
1/8*1000 = 125 (5730*3= 17190 years to decay)

Second question part b, answer should be 22.5million but mine was 24.
Working:
t = 0 (5m)
t = 1 (5+1 = 6m)
t = 2 (3+6 = 9m)
t = 3 (9+4 = 13m)
t = 4 (13+5 = 18m)
t = 5 (6+ 18= 24m)

P.S. the equation for part a that i found out from ur explanation and step was r = t+ 1

You might ask your instructor if the answer provided for the first question perhaps contains a typo, since reversing the 0 and the 9 would give your answer.