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You are exploring a hidden cave when suddenly, your flashlight goes out. Luckily, you have a 10-inch wax candle with you... Whew! You light the candle, but you must get out of the cave quickly, because every two minutes, one inch of the wax candle melts. Also, once the height of the candle reaches 2 inches or less, you will no longer be able to hold it(otherwise you would be burned). How long do you have to get out of the cave safely?

That is the word problem. I'm having some trouble coming up with an equation to it. Can anyone tell me what the algebraic representation of this function would be? This is part of my homework that is due 10/23/13.

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    $\begingroup$ What have you tried so far? What troubles are you running into in setting up an equation to solve? $\endgroup$
    – Dan
    Oct 22, 2013 at 23:29
  • $\begingroup$ @Dan I just have No idea how to solve it. I've tried f(x)=2x-1, but I realized that wouldn't work. $\endgroup$ Oct 22, 2013 at 23:30
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    $\begingroup$ "Tried f(x)=2x-1" doesn't really mean anything unless you know what aspects of the problem $f(x)$ and $x$ are supposed to represent. $\endgroup$
    – Jack M
    Oct 22, 2013 at 23:38
  • $\begingroup$ All I want to know what the function is, algebraically. f(x)=mx+b. $\endgroup$ Oct 22, 2013 at 23:40
  • $\begingroup$ So, what is m, and what is b? (the y-intercept?) $\endgroup$ Oct 22, 2013 at 23:40

2 Answers 2

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Go through the word problem carefully, noting each piece of (possibly-) relevant information at a time. Then try to build up a function/equation/whatever you need from the information.

For example, in this problem you might make the following quick observations: - The candle is $10$ inches tall. - The length of the candle decreases by $1$ inch every $2$ minutes - You're trying to figure out how long until the candle is $2$ inches tall (i.e., how long you have until you can no longer hold the candle).

You could definitely come up with an equation to solve here algebraically, but in this case it's probably going to be easier for you to just make a table of values giving the length of the candle after it's been lit for $t$ minutes (where, say, $t = 2, 4, 6, \ldots$). How many minutes until the candle is $2$ inches long?

Edit: If you're instructed to come up with a function $f(t)$ for the length (in inches) of the candle $t$ minutes after being lit, first assume that $f(t)$ is linear (which is reasonable here). Then you know $f(0) = 10$. You can compute $f(2)$ from the above observations. Now you have two points $(0,10)$, $(2,f(2))$ on this line, and so you can solve for the general formula.

(Since it's worth being nitpicky here, you should really have a piecewise function if you want to define $f(t)$ for all $t \geq 0$ (since otherwise you'd be saying that eventually the length of the candle is negative. But that's a secondary concern.)

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  • $\begingroup$ But is t the y-intercept or the slope? $\endgroup$ Oct 22, 2013 at 23:42
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    $\begingroup$ @JerryRox Neither. It may help if you forget about what you learned in class for a little while and just try to solve this using common sense. $\endgroup$
    – Jack M
    Oct 22, 2013 at 23:43
  • $\begingroup$ How is it neither? If I want to turn this verbal representation into an algebraic one, I need to have a slope and a y-intercept. $\endgroup$ Oct 22, 2013 at 23:45
  • $\begingroup$ Oh my gosh. Thank you sooo much. $\endgroup$ Oct 22, 2013 at 23:48
  • $\begingroup$ Not a problem. Hope this helps give you a better idea of how to think through these problems! :) $\endgroup$
    – Dan
    Oct 22, 2013 at 23:53
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The algebraic representation of what function? There's your starting point.

What are the quantities involved?

  • The amount of time in minutes since you lit the candle: call that $t$.
  • The height of the candle in inches: call the h.

Now, you need to figure out what $t$ will be when $h$ is equal to $0$, so why not start by trying to get a formula that tells you the value of $h$ based on the value of $t$?

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