Precalculus math question

Question

The ratio of carbon 14 to carbon 12 in a Piece of wood discovered in a cave is R=(1/8)¹⁴. Estimate the age of the piece of wood using the carbon dating model R=1/10^12 *e^-t/8223, where R represent the ratio of carbon, and t is time in year .

Answer 1

Filling in for R, the equation becomes

(1/8)¹⁴=1/10¹² *e^-t/8223

First we can divide by 1/10¹² to get:

10¹²/8¹⁴ = e^-t/8223

By raising both to 8223rd power, it becomes:

(10¹²/8¹⁴)⁸²²³ = e^-t

By finding the natural logarithm of both sides, we get:

ln((10¹²/8¹⁴)⁸²²³) = -t

Since the natural log is being raised to a power, we can multiply it by that exponent, making:

8223 * ln(10¹²/8¹⁴) = -t

Solving for this, we get

t = 12,180 years