SOLUTION: Hi, I need help on my word problem homework. Would someone kindly help me please.Please show me the steps on how to solve the problem as well. Thank you. Between 1989 and 1998,

Algebra ->  Functions -> SOLUTION: Hi, I need help on my word problem homework. Would someone kindly help me please.Please show me the steps on how to solve the problem as well. Thank you. Between 1989 and 1998,       Log On


   

Question 919253: Hi, I need help on my word problem homework. Would someone kindly help me please.Please show me the steps on how to solve the problem as well. Thank you.
Between 1989 and 1998, the population of small town, USA (in thousands) can be modeled by f(x)=0.8x^2-12.8x+54.2, where x=0 represents 1989. Based on this model, in what year did the population of small town reach its minimum?
My teacher said that the answer is: 1997
Would you please show me how he got the answer. Than you.
Found 2 solutions by KMST, jim_thompson5910:
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The population is given by f%28x%29=0.8x%5E2-12.8x%2B54.2 with x= number of years after 1989.
The maximum/minimum for a quadratic function g%28x%29=ax%5E2%2Bbx%2Bc happens for x=-b%2F2a .
(It is a minimum if a%3E0, a maximum if a%3C0 .
In this case, a=0.8 and b=-12.8 ,
so there is a minimum forx=-%28-12.8%29%2F%282%2A0.8%29=12.8%2F1.6=8 ,
and x=8 corresponds to 1989%2B8=1997 .

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
In general, the min of y+=+ax%5E2+%2B+bx+%2B+c is when x+=+-b%2F%282a%29. This is assuming a+%3E+0. If a+%3C+0, we'll have a max and not a min.

f%28x%29=0.8x%5E2-12.8x%2B54.2 is in that form where a+=+0.8 and b+=+-12.8. We don't need c in this case.

Plug the values in:

x+=+-b%2F%282a%29

x+=+-%28-12.8%29%2F%282%2A0.8%29

x+=+8

Recall that x = 0 represents 1989. So x = 1 represents 1989+1 = 1990, etc etc

x = 8 means 1989 + 8 = 1997

So 8 years after the starting point of 1989, ie in the year 1997, the population is at the minimum.

If you want to know this min population value, plug it back into f(x) and evaluate.

Let me know if you need more help or if you need me to explain a step in more detail.
Feel free to email me at jim_thompson5910@hotmail.com
or you can visit my website here: http://www.freewebs.com/jimthompson5910/home.html

Thanks,

Jim