A candle burns down at the rate of 0.5 inches per hour. The original height of the candle was 8 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, 8) would represent a height of 8 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. (2 points)
Part C: If the rate at which the candle burned was 0.4 inches per hour instead of 0.5 inches per hour, would the relation be a function? Explain your answer using input and output values. (3 points)
Part A:
To obtain the ordered pairs, we need to determine the height of the candle at different points in time. Since the candle burns down at a rate of 0.5 inches per hour, we can subtract 0.5 inches from the previous height for each hour that passes.
Starting with the original height of 8 inches:
(0, 8)
After 1 hour, the height would be 8 - 0.5 = 7.5 inches:
(1, 7.5)
After 2 hours, the height would be 7.5 - 0.5 = 7 inches:
(2, 7)
Continuing this pattern, we get the following ordered pairs:
(0, 8)
(1, 7.5)
(2, 7)
(3, 6.5)
(4, 6)
(5, 5.5)
Part B:
Yes, this relation is a function. In a function, each input value (x) corresponds to exactly one output value (y). If we look at the ordered pairs we created, each time value (x) has a unique height value (y). For example, 1 hour corresponds to a height of 7.5 inches, and this height is not repeated for any other time value. Therefore, this relation is a function.
Part C:
If the rate at which the candle burned was 0.4 inches per hour instead of 0.5 inches per hour, the relation would still be a function. The change in burning rate does not affect the fact that each input value (x) corresponds to exactly one output value (y). The specific input-output pairs might change, but the fundamental nature of the relation as a function remains the same.