Question

From 1990 to 2000, the population of Florida increased by 2.13% per year. The population (in millions) in the year 1990 was 16 Estimate the population in the year 1997 (in millions). Round to the nearest integer

19 million
18 million
62 million
15 million
​

Answer 1

19 million

Explanation:

So generally whenever we express a exponential function, it's in the form:
`y=a(b)^x`

where a=initial value aka when x=0

in this case, b can be expressed as:
`1+r`, where r is the growth rate, which in this case is 2.13%, and expressing that as a decimal (divide by 100), r=0.0213, thus b=1.0213

This gives us the following equation:
`y=16(1.0213)^x`

To kind of give you a intuitive reason as to why this makes sense, think of that 1.0213 wrapping around the previous value.

So when x=0, it's actually just 16, since any number (besides 0) to the power of 0, equals 1.

When x=1, we get:
`1.0213(16)`, which can be expressed as a 2.13% increase from year 0

When x=2, we get:
`1.0213(1.0213(16))`, which can be expressed as a 2.13% increase from year 1

when x=3, we get:
`1.0213(1.0213(1.0213(16)))`, which can be expressed as a 2.13% increase from year 2.

In other words, each year we're getting another 1.0213 to multiply by (very definition of an exponent).

So to calculate the population in 1997, we put in x=7, since it's 7 years after 1990

We get the following equation:
`y=16(1.0213)^7`

Using a calculator this simplifies to approximately:
`18.5436`

Rounding this to the nearest integer would yield 19, and since the values are expressed in millions, the answer would be 19 million