Question

the formula ( a = 82.4e^{-0.002t} ) models the population of germany, a, in millions, t years after 2006.

1. what was the population of the county in 2006?
2. when will the population of germany reach 81.5 million?

Explanation:

Step1: Substitute \(A = 81.5\) into the formula

Given \(A=82.4e^{- 0.002t}\), when \(A = 81.5\), we have \(81.5=82.4e^{-0.002t}\).

Step2: Solve for \(e^{-0.002t}\)

Divide both sides by \(82.4\): \(\frac{81.5}{82.4}=e^{-0.002t}\), so \(e^{-0.002t}\approx0.989078\).

Step3: Take the natural - logarithm of both sides

Using the property \(\ln(e^{x})=x\), we get \(\ln(0.989078)=- 0.002t\).
Since \(\ln(0.989078)\approx - 0.011\), then \(-0.011=-0.002t\).

Step4: Solve for \(t\)

Divide both sides by \(-0.002\): \(t=\frac{-0.011}{-0.002}=5.5\)

Answer:

\(t = 5.5\) years. So the population will reach \(81.5\) million \(5.5\) years after \(2006\), that is in \(2011.5\) (mid - 2011).