Use the table to answer the question.
Hours | Miles walked
14 | 1
12 | 2
34 | 3
1 | 4
It takes Julian 12 hour to walk 2 miles. He decides to start walking in his spare time, but because he wants to make sure he has enough time to go a desired distance he created the data table. At which ratio does the constant of proportionality appear?
A (1, 4)
B (14, 1)
C (1, 14)
D (4, 1)
The constant of proportionality represents the relationship between the hours and miles walked. To find the constant of proportionality, we need to identify a ratio where the number of hours and the number of miles form a consistent relationship.
Looking at the table, we can see that when Julian walks for 14 hours, he walks 1 mile. This means that the ratio of hours to miles is 14:1.
Therefore, the ratio where the constant of proportionality appears is (14, 1).
Answer: A (14, 1)
To determine the ratio at which the constant of proportionality appears, we need to look for a consistent relationship between the hours and miles walked. In this case, we can calculate the rate of change (slope) between the hours and miles.
To find the rate of change, we can divide the change in miles by the corresponding change in hours. Let's calculate the rates for each pair of data points:
1. Between (14, 1) and (12, 2):
rate = (2 - 1) / (12 - 14) = 1 / -2 = -1/2
2. Between (12, 2) and (34, 3):
rate = (3 - 2) / (34 - 12) = 1 / 22
3. Between (34, 3) and (1, 4):
rate = (4 - 3) / (1 - 34) = 1 / -33 = -1/33
From these calculations, we can see that the rate of change is consistent and equal to -1/2. Therefore, the answer is (A) (1, 4).
To determine the constant of proportionality in this table, we need to check if there is a consistent ratio between the hours and miles walked for each row.
Let's calculate the ratio for each row:
For the first row (14 hours, 1 mile):
Ratio = 14 hours / 1 mile = 14/1 = 14
For the second row (12 hours, 2 miles):
Ratio = 12 hours / 2 miles = 12/2 = 6
For the third row (34 hours, 3 miles):
Ratio = 34 hours / 3 miles = 34/3 = 11.33
For the fourth row (1 hour, 4 miles):
Ratio = 1 hour / 4 miles = 1/4 = 0.25
From these calculations, we can see that the only pair with a consistent ratio is (1, 4). Therefore, the constant of proportionality appears at option A, which is (1, 4).