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Determine the rational function that gives the annual cost of a security system

  • #1

Homework Statement



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(x) 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 $200 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(x) (2000+108t)/t..right? Am i supposed to subtract the two functions?
 

Answers and Replies

  • #2
jbunniii
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let t be the time in year

$9/month = $108/year

C(t)= 1500+108t

let A(x) be the average annual cost of the system

A(t)= (1500+108t)/t, t<8
OK, this looks good. (But if your time variable is t, then stick with that and don't switch it to x. It just confuses the reader.)

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?
Yes, that's right. (Well, at t = 0, not x = 0.) It makes sense, doesn't it? Even if you buy the thing and only operate it for a very short period of time, you still spent $1500, and you can only average that cost over a short time period, so your average cost over the time interval must be very high in that case, right?

For example, if you only used it for one second, you would pay $1500 plus some tiny amount due to the operating cost, so just call it $1500. But that's $1500 for one second of use, which is a rate of

(1500 dollars / second) x (3600 seconds / hour) x (8760 hours / year) = $47,304,000,000/year !!

c)If another brand of security system costs $200 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(x) (2000+108t)/t..right? Am i supposed to subtract the two functions?
I assume you mean it costs $2000, not $200.

It depends on how long you use it, right? See if you can calculate the "crossover" value of t, meaning, if you use it for less time than that, then system A is a better deal, whereas if you use it for longer than that, then system B is a better deal.
 
  • #3
OK, this looks good. (But if your time variable is t, then stick with that and don't switch it to x. It just confuses the reader.)



Yes, that's right. (Well, at t = 0, not x = 0.) It makes sense, doesn't it? Even if you buy the thing and only operate it for a very short period of time, you still spent $1500, and you can only average that cost over a short time period, so your average cost over the time interval must be very high in that case, right?

For example, if you only used it for one second, you would pay $1500 plus some tiny amount due to the operating cost, so just call it $1500. But that's $1500 for one second of use, which is a rate of

(1500 dollars / second) x (3600 seconds / hour) x (8760 hours / year) = $47,304,000,000/year !!



I assume you mean it costs $2000, not $200.

It depends on how long you use it, right? See if you can calculate the "crossover" value of t, meaning, if you use it for less time than that, then system A is a better deal, whereas if you use it for longer than that, then system B is a better deal.
Sorry, for part C it is actually $2000

(108(8years)+150)/8 = 295.5

(108(12 years) + 150)/12 = 274.66

The 2nd system has a lower average yearly cost over it's lifetime

Correct?
 
Last edited:

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