The number of airline passengers in 1990 was 466 million. The number of passengers traveling by airplane each year has increased exponentially according to the model, P ( t ) = 466 ⋅ 1.035 t , where t t is the number of years since 1990. In what year is it predicted that 900 million passengers will travel by airline?

We have been given an exponential growth formula [tex]P(t)=466\cdot 1.035^t[/tex], which represents number of passengers traveling by airplane since 1990.

To find the year in which 900 million passengers will travel by airline, we will equate the given formula by 900 and solve for t as:

[tex]900=466\cdot 1.035^t[/tex]

[tex]\frac{900}{466}=\frac{466\cdot 1.035^t}{466}[/tex][tex]1.9313304721030043=1.035^t[/tex]

Take natural log of both sides:

[tex]\text{ln}(1.9313304721030043)=\text{ln}(1.035^t)[/tex]

Using property [tex]\text{ln}(a^b)=b\cdot \text{ln}(a)[/tex], we will get:

[tex]\text{ln}(1.9313304721030043)=t\cdot \text{ln}(1.035)[/tex]

[tex]0.658209129198=t\cdot 0.034401426717[/tex]

[tex]\frac{0.658209129198}{0.034401426717}=\frac{t\cdot 0.034401426717}{0.034401426717}[/tex]

[tex]19.1331927775=t\\\\t=19.1331927775[/tex]

This means that in the 20th year since 1990, 900 million passengers would travel by airline.

[tex]1990+20=2010[/tex]

Therefore, 900 million passengers would travel by airline in 2010.