Question

The exponential model A=58.7e^0.02t describes the population,A, of a country in millions, t years after 2003. Use the model to determine when the population of the country will be 89 million.

Answer 1

About 20.81 years

Explanation:

89 million is the "final population" -- population after t years.

So, 89 million would be in "A" in the equation. Then we will have to solve for "t" by taking LN (natural logarithm). That is how we solve exponential equations.

So,

`A=58.7e^(0.02t)\\89=58.7e^(0.02t)\\(89)/(58.7)=e^(0.02t)\\1.5162=e^(0.02t)`

Now we recognize the exponential rule of:

Ln(e) = 1

and we use the property:

`Ln(a^b)=bLn(a)`

Now, we solve by taking Ln of both sides:

`1.5162=e^(0.02t)\\Ln(1.5162)=Ln(e^(0.02t))\\Ln(1.5162)=0.02tLn(e)\\Ln(1.5162)=0.02t\\t=(Ln(1.5162))/(0.02)\\t=20.81`

So, population would be 89 million in about 20.81 years