A candle burns down at the rate of 0.5 inches per hour. The original height of the candle was 6 inches.
Part A: Write a list of 6 ordered pairs to show the height of the candle in inches (y) as a function of time in hours (x) from the first hour after it started burning. For example, the point (0, 6) would represent a height of 6 inches after 0 hours. Explain how you obtained the ordered pairs. (5 points)
Part B: Is this relation a function? Justify your answer using the list of ordered pairs you created in Part A.
Part C: If the rate at which the candle burnt was 0.3 inches per
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A bit of Part C of the question is missing I think!....
In relation to the first two parts though:
Part A: The question goes through what form our six ordered pairs should be. The best thing to do here is to probably imagine that we're plotting these points on a graph, with the candle height (in inches) on the y-axis and our time (in hours) on the x-axis. So, we're plotting the decrease in the candle's height as it burns over time.
The question tells us that this decrease is constant (i.e. it occurs at a constant rate of 0.5 inches per hour of burning). That means for every 1 hour period we have the candle burning for (no matter how long we had it burning previously), it's height will go down by 0.5 inches.
We're told what the first point, or ordered pair, (x, y) would be for the candle at 0 hours: (0, 6), with 6 inches being it's original height. Thus, our next pair would be for after 1 hour (x = 1), but now the candle's height has gone down by 0.5 inches (y = 5.5). This gives us our point (1, 5.5) and we continue this on five more times until we are left with six ordered pairs of points for the candle's height at each time interval.
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omg thank you so much @Ciarán95
Part B: If we were to plot our our ordered pairs of values (our points) that we identified in Part A and we were to join them up, we would we that they would form a nice straight line on the graph. This results from the linear relationship which exists between the time we have the candle burning for and its height, as the rate at which it decreases is constant throughout and never fluctuates. We can define this line using a particular equation in terms of x and y - you may have seen equations like these before.
In maths, a function is like a 'machine' or a system from which we can input a certain value and we get out a certain value. There's a few different rules as to how we go about defining whether something is a function or not, but any system which matches one input to exactly one particular output will be a valid function.
The equation of our line is like a 'system' or a 'machine' in this case - we could use it to predict the candles height (the y-value) for a given time values we input (x-value), other that those which we already know at the 1-hour intervals. Similarly, if we know the candle's height to be a particular value (y-value), we can input this into our equation and use it to find out how long the candle had been burning for it to reach that height (the corresponding x-value).
That's just basically a general view of how you would go about answering Part B. I'm not 100% sure if this is the way of specifically going about showing it in relation to the ordered pairs from part A - you could plot the pairs/points on a graph and actually work out the equation of the line joining them together, as I was talking about earlier, but that mightn't be required.
If you want some more info on how we go about defining a function in general, this like might be of interest to you:
Anyway, hope that helped you out a bit @na1501 ! :)