Question

The half life of the radioactive isotope carbon 14 is 5730 years. suppose we start with n milligrams of carbon 14. a) find a(t) the amount of carbon 14 after t years. b) what percentage of the original amount of carbon 14 is left after 20,000 years? c) if an old wooden tool is found in a cave and the amount of carbon 14 present in it is estimated to be only 42% of the original amount, how old is the tool? justify your answer.

a) Determine a(t), the amount of carbon 14 after t years.

b) Calculate the percentage of the original amount of carbon 14 remaining after 20,000 years.

c) If a wooden tool is discovered in a cave with only 42% of its original carbon 14, estimate the tool's age and provide justification.

Answer 1

To find the amount of carbon-14 after t years, use the formula a(t) = n * (1/2)^(t/5730). To find the percentage of the original amount of carbon-14 remaining after 20,000 years, divide a(20,000) by n and multiply by 100. To find the age of a wooden tool with 42% of its original carbon-14, substitute a(t) = 0.42n into the formula and solve for t.

Step-by-step explanation:

The amount of carbon-14 after t years can be calculated using the formula:

a(t) = n * (1/2)^(t/5730)

Where n is the starting amount of carbon-14 and t is the number of years.

To calculate the percentage of the original amount of carbon-14 remaining after 20,000 years, substitute t = 20,000 into the formula to get a(20,000). Divide a(20,000) by n and multiply by 100 to get the percentage.

If a wooden tool is found with only 42% of its original carbon-14, you can use the formula to find the age of the tool. Substitute a(t) = 0.42n into the formula and solve for t to get the age of the tool.