A candle burns down at the rate of 0.5 inches per hour. The original height of the candle was 10 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, 10) would represent a height of 10 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 can use the formula: height (y) = original height - rate of burning (0.5 inches/hour) * time (x).
The first hour after it started burning is represented by x = 1, so we can substitute it into the formula:
y = 10 - (0.5 * 1) = 9.5
So, the first ordered pair is (1, 9.5).
Similarly, we can calculate the rest of the ordered pairs:
(2, 9) - after 2 hours
(3, 8.5) - after 3 hours
(4, 8) - after 4 hours
(5, 7.5) - after 5 hours
(6, 7) - after 6 hours
Part B:
Yes, this relation is a function. A function is a relation between two sets where each input is associated with exactly one output. In this case, the ordered pairs in Part A shows that for every hour (input), there is only one corresponding height (output).
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 reason is that the relation being a function is determined by each input having exactly one output, not by the specific values of the inputs or outputs. Therefore, even if the rate changes, as long as each hour (input) only corresponds to one height (output), the relation will still be a function.