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In 2000, the population of Massachusetts
was 6.3 million people and was growing at a rate of
about 0.32% per year. At this growth rate, the function
ƒ(x) = 6.3(1.0032)^x gives the population, in millions x
years after 2000. Using this model, find the year when
the population reaches 7 million people.

Respuesta :

Answer:

The population reaches 7 million people in the year 2032.

Step-by-step explanation:

We have that

[tex]f(x) = 6.3*(1.0032)^{x}[/tex]

Using this model, find the year when the population reaches 7 million people.

This is x years after 2000, in which x is found when f(x) = 7. So

[tex]f(x) = 6.3*(1.0032)^{x}[/tex]

[tex]7 = 6.3*(1.0032)^{x}[/tex]

[tex](1.0032)^{x} = \frac{7}{6.3}[/tex]

[tex](1.0032)^{x} = 1.11[/tex]

We have to following logarithm rule

[tex]\log{a^{x}} = x\log{a}[/tex]

So we apply log to both sides of the equality

[tex]\log{(1.0032)^{x}} = \log{1.11}[/tex]

[tex]x\log{1.0032} = \log{1.11}[/tex]

[tex]x = \frac{\log{1.11}}{\log{1.0032}}[/tex]

[tex]x = 32.66[/tex]

2000 + 32.66 = 2032

The population reaches 7 million people in the year 2032.