Suppose an Egyptian mummy is discovered in which the amount of carbon-14 present is only about two-fifth years is given by \( y=y_{0} e^{-0.0001216 t} \), where \( y_{0} \) is the amount of carbon-14 present in living human beings. At 11 of 13 , About \( \square \) yr ago the Egyptian had died. (Round to the nearest integer as needed.)

To find out how many years ago the Egyptian died, we first need to determine the value of \( t \) when the amount of carbon-14 remaining is two-fifths of the original amount, \( y_0 \). Given that: \[ y = y_0 e^{-0.0001216 t} \] Substituting \( y = \frac{2}{5} y_0 \): \[ \frac{2}{5} y_0 = y_0 e^{-0.0001216 t} \] Dividing both sides by \( y_0 \) (assuming \( y_0 \) is not zero): \[ \frac{2}{5} = e^{-0.0001216 t} \] To solve for \( t \), we take the natural logarithm of both sides: \[ \ln\left(\frac{2}{5}\right) = -0.0001216 t \] Now we can solve for \( t \): \[ t = \frac{\ln\left(\frac{2}{5}\right)}{-0.0001216} \] Calculating \( \ln\left(\frac{2}{5}\right) \): \[ \ln\left(\frac{2}{5}\right) \approx -0.714 \] Substituting the value into the equation: \[ t \approx \frac{-0.714}{-0.0001216} \approx 5877.3 \] Rounding to the nearest integer, we find that the Egyptian had died about \( 5877 \) years ago.