You have 5 grams of carbon-14; whose half-life is 5730 years.

a)Write the rule of the function that gives the amount of carbon-14 remaining after x years.
b)How much carbon-14 will be left after 4,000 years?
c)When will there be just 1 gram left?

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a) The rule of the function that gives the amount of carbon-14 remaining after x years can be expressed using the formula:

N(t) = N0 * (1/2)^(t/h)

Where:
- N(t) is the amount of carbon-14 remaining after t years.
- N0 is the initial amount of carbon-14 (5 grams, in this case).
- (1/2) is the decay constant, representing the fraction of carbon-14 that remains after one half-life.
- t is the number of years that have passed.
- h is the half-life of carbon-14 (5730 years, in this case).

b) To find out how much carbon-14 will be left after 4,000 years, we can substitute t = 4000 into the equation:

N(t) = N0 * (1/2)^(t/h)
N(4000) = 5 * (1/2)^(4000/5730)

Using a calculator, we can evaluate this expression to find the amount of carbon-14 remaining after 4,000 years.

c) To determine when there will be just 1 gram of carbon-14 left, we need to solve the equation:

1 = 5 * (1/2)^(t/5730)

We can rearrange this equation and solve for t. For this, we can take logarithms to simplify the calculation:

log(1) = log[5 * (1/2)^(t/5730)]

Since log(1) equals zero, we can simplify further:

0 = log[5 * (1/2)^(t/5730)]

Now, divide both sides of the equation by log(5) to isolate the exponential term:

0 = (t/5730) * log(1/2)

To solve for t, divide both sides by log(1/2):

0 / log(1/2) = t / 5730

Therefore, t = 5730 * (0 / log(1/2)).

Note: For this particular equation, since log(1/2) is negative, the solution for t will yield a negative value that is not meaningful in this context. It indicates that there will never be precisely 1 gram of carbon-14 remaining. However, if we round the result to the nearest whole number, we can obtain an estimated time when the amount of carbon-14 remaining will be close to 1 gram.