Please help me answer the following question. I am particularly stuck on #3. Thanks!Michigan’s population is declining at a rate of 0.5% per...

Michigan’s population is declining at a rate of
0.5% per year. In 2004, the state had a population of
10,112,620.

1. Write a function to
express this situation.

In 2004 the population
was 10,112,620
In 2005 the population was
(10,112,620)(1-.005)=10,062,057
In 2006 the population was
(10,112,620)(1-.005)(1-.005)
etc...

Each year the
population is multiplied by .995 to compute the following years
population.

Thus the function is
`f(x)=(10,112,620)(.995)^t` where t is the number of years since
2004.

2. If this rate continues, what will the
population be in 2012?

Using the function from
(1) and `t=2012-2004=8` we get
Population = `(10,112,620)(.995)^8=9,715,124`

3. When will the population of
Michigan reach 9,900,000?

If this is a middle
school assignment, you could build a table or use a spreadsheet to estimate t between
4.2 and 4.3 years. To get the exact answer we use
logarithms:

`10,112,620(.995)^t = 9,900,000`
`t
log(.995)=log((9,900,000)/(10,112,620))`
` t ~~ 4.239
`

4. What was the population in year 2000,
according to this model?

Here, t=-4 so plugging
in we get t=10,317,426.