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...

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?

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

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.