3. While an organism is living, it is continually changing its carbon with the environment. (Plants directly absorb CO2 from the atmosphere, while animals get their carbon from living plants.) Gamma radiation that bombards the earth keeps the ratio of carbon-14 to carbon-12 fairly constant in the atmospheric CO2. The 14C that entered a living organism can be used to determine how much time has passed since it died. It has been determined that radioactive carbon, 14C, decays with a half-life of 5730 years, i.e., in 5730 years the quantity of 14C decreases to half its original amount. Living tissue shows a radioactivity of about 15.3 disintegrations per minute (dpm) per gram of carbon, which means that 5730 years after the organism has died, it will exhibit only half or 7.65 dpm per gm of carbon. Let R(t) represent the number of dpm per gram of carbon from an ancient object. The amount of 14C remaining at any time t is proportional to R(t) and can be determined by the initial value problem:

R'(t) = -kR(t), R(0) = R0,

where R0 = 15.3 is the initial number of dpm per gram of carbon.

a. Solve this initial value problem and use the information on the half-life of 14C above to find k to 4 significant figures. Graph the solution for 0 < t < 20,000.

b. Find the age of a fossilized bone from a man in Western Pennsylvania that contains 16% of its original 14C. When did a Kenyan man die if his bone contains 8% of its original 14C?

c. Estimate the age of each object in the table below if the readings were taken in 1950:

Object

dpm per gram

Chair Leg from tomb of Tutankhamen

10.14

House Beam from Babylon (Hammurabi era)

9.52

Giant Sloth Dung (Gypsum Cave in Nevada)

4.17

 

d. Suppose the error in each of the readings above is ±0.3 dpm. Find the range of possible ages for each of the objects.

e. Estimate the measured disintegrations per minute per gram of carbon for each of the following objects. What would be the error in the dpm readings?

Object

Age in Years

Charcoal from the Lascaux caves in France

15,516±900

Linen wrappings from the Dead Sea Scrolls

1,917±200

 

Charcoal from Stonehenge in England

3,798±275