Ann went on a 10 hour bicycle trip. The graph models the relationship between the time and the total distance traveled by Ann. The table shows selected points on the graph. T=0,1,4,7,10 d(t)=0,3,5,5,10 draw a graph of anns speed in mph in terms of the time in hours.
between 0 and 1 hour Ann went 3 miles per hour
between 1 hour and 4 hours, Ann went 2 miles (5-3) so 2 miles /3 hours = .67 mph
between 4 hours and 7 hours Ann went zero miles (5-5) so zero miles/hour (lunch break?)
between 7 ours and 10 hours (3 hours) Ann went 5 miles (10-5) so 5/3 = 1.67 mph
To draw a graph of Ann's speed in mph in terms of time in hours, we need to find the rate at which her distance traveled changes over time. This can be calculated by finding the slope of the line connecting each pair of points on the graph.
Using the provided points on the graph (0,0), (1,3), (4,5), (7,5), and (10,10), we can calculate the speed at each time point by dividing the change in distance by the change in time.
Speed (mph) = Change in distance / Change in time
For the given points, the calculated speeds are as follows:
At t = 0 hours to t = 1 hour:
Speed (mph) = (3 - 0) / (1 - 0) = 3 mph
At t = 1 hour to t = 4 hours:
Speed (mph) = (5 - 3) / (4 - 1) = 2/3 mph
At t = 4 hours to t = 7 hours:
Speed (mph) = (5 - 5) / (7 - 4) = 0 mph
At t = 7 hours to t = 10 hours:
Speed (mph) = (10 - 5) / (10 - 7) = 5/3 mph
Based on these calculations, we can plot the four speeds on the graph, using the corresponding time points.
The graph would look like this:
^
|
|
Speed (mph)| 3
| /
| /
| /
+-----------------
1 4 7 10 Time (hours)
Note that the speed is constant at 3 mph between 0 and 1 hour, then decreases gradually to 0 mph between 4 and 7 hours, and finally increases to 5/3 mph between 7 and 10 hours.
To draw a graph of Ann's speed in mph in terms of time in hours, we need to calculate the speed at each point by finding the rate of change of distance with respect to time.
First, we need to calculate the average speed between each pair of points on the graph. We can do this by dividing the change in distance by the change in time.
Let's calculate the average speed between the points T=0 and T=1:
- d(1) - d(0) = 3 - 0 = 3 miles
- T(1) - T(0) = 1 - 0 = 1 hour
So, the average speed between T=0 and T=1 is 3 miles per 1 hour, or 3 mph.
Similarly, we can calculate the average speeds between the other pairs of points:
- Between T=1 and T=4: average speed = (5 - 3) miles / (4 - 1) hours = 2 miles per 3 hours, or 0.67 mph.
- Between T=4 and T=7: average speed = (5 - 5) miles / (7 - 4) hours = 0 miles per 3 hours, or 0 mph.
- Between T=7 and T=10: average speed = (10 - 5) miles / (10 - 7) hours = 5 miles per 3 hours, or 1.67 mph.
Now, we can plot these average speeds on the graph. We'll use the time values on the x-axis and the average speed values on the y-axis.
The graph should look like this:
Speed (mph)
^
|
3 +
|
|
|
2 +
|
|
| 1.67 +
1 +
|
|
0.67 +
|
|
|
0 +_____________________
0 1 4 7 10 Time (hours)
Note that for the intervals where the speed is 0 mph (between T=4 and T=7), the graph should have horizontal lines.
This graph represents Ann's speed in mph in terms of the time in hours.