Jim runs laps around his block every morning before school. The graph shows a proportional relationship between time in minutes and the number of laps completed.
Identify the constant of proportionality from the graph.
A: 2
B: 3
C: 2/3
D: 3/2
(Graph: A - 2 Laps, 3 Minutes; B - 4 Laps, 6 Minutes; C - 6 Laps, 9 Minutes.)
To find the constant of proportionality, we need to determine the ratio of laps to time for any point on the graph. We can choose any point, but it is easiest to choose point A, which has coordinates (3 minutes, 2 laps). The ratio of laps to time is 2/3.
Therefore, the constant of proportionality from the graph is C: 2/3.
Explain.
To find the constant of proportionality, we need to determine the ratio of the dependent variable (in this case, the number of laps completed) to the independent variable (in this case, time in minutes). The constant of proportionality represents how the dependent variable changes in relation to the independent variable.
Looking at the graph, we can observe that as the time increases, the number of laps completed also increases. We can see that for every 3 minutes, Jim completes 2 laps (point A), for every 6 minutes he completes 4 laps (point B), and for every 9 minutes he completes 6 laps (point C).
Since the ratio of laps to time remains consistent, we can conclude that the constant of proportionality is the same for all points on the graph. The ratio is 2 laps / 3 minutes, which can be expressed as 2/3.
Therefore, the constant of proportionality from the graph is C: 2/3.
Choose the equations which have a greater rate of change than the graph.
(Graph: 6 miles, 20 Minutes; 12 miles, 40 minutes; 18 miles, 60 minutes; 24 miles, 80 minutes; 30 miles, 100 minutes)
A: y = 1/4x
B: y 1/3x
C: y = 3/10x
D: y = 2/5x
To determine which equations have a greater rate of change than the given graph, we need to compare the slopes of the equations to the slope of the graph.
The slope of a line represents the rate of change or how much the dependent variable (y) changes for every unit increase in the independent variable (x).
Looking at the graph, we can see that for every 20 minutes, the number of miles traveled increases by 6 miles. Therefore, the slope of the graph is 6 miles / 20 minutes, which simplifies to 3/10.
Now, let's analyze each equation:
A: y = 1/4x
The slope of this equation is 1/4. This is less than the slope of the graph (3/10). Therefore, equation A does not have a greater rate of change than the graph.
B: y = 1/3x
The slope of this equation is 1/3. This is less than the slope of the graph (3/10). Therefore, equation B does not have a greater rate of change than the graph.
C: y = 3/10x
The slope of this equation is 3/10, which is exactly the same as the slope of the graph. Therefore, equation C has the same rate of change as the graph, but not a greater rate of change.
D: y = 2/5x
The slope of this equation is 2/5. This is greater than the slope of the graph (3/10). Therefore, equation D does have a greater rate of change than the graph.
In conclusion, equation D (y = 2/5x) has a greater rate of change than the given graph.
To determine the constant of proportionality, we need to find the ratio of laps to minutes that remains the same throughout the graph.
Looking at the graph, we see that when the time is 3 minutes, the number of laps completed is 2.
Similarly, when the time is 6 minutes, the number of laps completed is 4.
To find the constant of proportionality, we can divide the number of laps by the time in minutes:
Constant of proportionality = Number of laps / Time in minutes
For the first set of data points (2 laps, 3 minutes), the constant of proportionality is:
2/3 = 0.6667
And for the second set of data points (4 laps, 6 minutes), the constant of proportionality is:
4/6 = 0.6667
As we can see, both sets of data points result in the same constant of proportionality, 0.6667 or 2/3.
Thus, the correct answer is C: 2/3.
To identify the constant of proportionality from the graph, we need to determine the ratio between the number of laps completed and the time in minutes for each data point.
Let's calculate the ratios:
For point A, the ratio is: 2 laps / 3 minutes = 2/3
For point B, the ratio is: 4 laps / 6 minutes = 4/6 = 2/3
For point C, the ratio is: 6 laps / 9 minutes = 6/9 = 2/3
As we can see, the ratios for all three points are the same. This means that the constant of proportionality is 2/3.
Therefore, the correct answer is C: 2/3.