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## Homework Statement

An antiques dealer has claimed that a tapestry is

**2000**years old having been loomed sometime in the first century BC. In order to determine the veracity of this claim a sample of Carbon-14 has been taken for dating. In a

**1g**sample,

**190ng**of Carbon-14 were present.

The half-life of Carbon-14 is

**5780 years**.

The below graph displays the amount of Carbon-14 present in the environment for the past 2100 years.

__Known variables:__

Current Carbon-14 present: 190ng

Half life of Carbon-14: 5780 years

__Wanted variables:__

Age of tapestry (t)

And original amount of Carbon-14 ([itex]N_{0}[/itex])

## Homework Equations

Taken from Wikipedia.

[itex]\frac{dN}{dt} = -\lambda N[/itex]

"...differential equation, where N is the number of radioactive atoms and λ is a positive number called the decay constant"

[itex]N(t) = N_{0}e^{-\lambda t}[/itex]

"...describes an exponential decay over a timespan t with an initial condition of N0 radioactive atoms at t = 0. Canonically, t is 0 when the decay started. In this case, N0 is the initial number of 14C atoms when the decay started."

## The Attempt at a Solution

First I calculated the decay constant of C14.

[itex]0.5 = e^{-\lambda 5780}[/itex]

[itex]\lambda = 1.209681344 x 10^{-4}[/itex]

Since the current amount of C14 is 190ng, the sample cannot be older than ~1550, since there was ~190ng of C14 in the environment at that time.

However, this leaves me with two unknowns, [itex]N_{0}[/itex] and [itex]t[/itex].

If I try to get the original amount of C14:

[itex]N(0) = N_{0}e^{-\lambda 0} = N_{0}e^{0} = N_{0}(1)[/itex]

[itex]N(0) = N_{0}[/itex]???

We've never done Carbon-14 dating before, and everything I've found so far has been from my own research. I feel as though I'm not understanding this correctly, and I'm unsure how to go about solving this.

Any and all guidance would be greatly appreciated.

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