A security system costs $9/month in costs (fixed cost). If a new security system costs $1500 to purchase and last for 8 years:
a)Determine the rational function that gives the annual cost of a security system as a function of the number of years you own the system. Using this function determine the average annual cost of the security system given above.[
Work Done
let C(t) be the cost function for the security system
let t be the time in year
$9/month = $108/year
C(t)= 1500+108t
let A(t) be the average annual cost of the system
A(t)= (1500+108t)/t, t<8
b)What are the asymptotes for this function and what do they mean in the context of the problem? Is it realistic?
The vertical asymptote is at x=0???? How come?
c)If another brand of security system costs $2000 but lasts 12 years, is this other brand of security sytsem worth the extra cost? Please justify your response.
Okay so,this function will be A(t) (2000+108t)/t..right? Am i supposed to subtract the two functions?
Why does "x = 0" turn up as an asymptope? Can you ever own something for 0-years? And that's not the only asymptope. You have a horizontal one.
Why would you subtract them? I applaud trying something, but don't just flail about for an answer. You have two functions, which one has the better cost over it's lifetime?Originally Posted by Bob
Thanks for responding,
Is my equation correct for part a though? A(t) = (1500+108t)/t? I substituted t=8 into the equation to get $295.50 as the average annual cost.
For part c, I believe subtracting both equations will get me the difference in cost. And that will help me determine which system is more economical?
2000+108t/t - 1500+108t/t = 500/t
meaning the 2nd system is more expensive
The equation in part A will determine the average cost over "t" years, yes.
Your question about which system is more cost effective needs to use this average. Think about what you have done in "English" not in "Math": if you subtract the average functions of one from the other, wont you just get the difference in the averages? The problem is, the time for both of these functions is NOT the same: one lasts 9 years, the other lasts 12 years. So if I said to you about an item in real life - "What is the better LONG TERM investment", wouldn't you evaluate each other their lifetimes?
Let me know what you get. Also see if you can interpret what the answer is in real world language, and not math-talk.
the 2nd system is better if one plans on using it for the entire 12 years. however, if one does not plan on committing to a longer term plan, the 1st plan is more economical because of its lower start-up fee. The 2nd system has a lower average yearly cost over its lifetime and is worth the extra cost if one commits to the 12 years.
Exactly. Sometimes it helps to put a problem (especially a word problem) in context by saying what it is asking (or what the solution is giving you) in "real world" words. If someone bought the 2000 dollar plan, they would save money in the long term, but would have a higher startup cost, and would be tied to that model for 12 years. Versus a lower startup cost, but a higher yearly average on the shorter term solution.
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