# 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 year. In 2004, the state had a population of...

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 year. In 2004, the state had a population of 10,112,620.*

* 1. Write a function to express this situation.*

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

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

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

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### 1 Answer

*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.