The Table Shows The Velocity Of A Bicyclist Riding For 30 Seconds. Use The Right Riemman Sum To Estimate The Distance Using 3 Intervals Of Length 10. Time (See) Velocity Ft Sec) 0 5 10 15 20 25 30 8 127 29B 222 170 Ft 630 It 200 Ft 170 Ft 230 Ft MacBook Air
The Correct Answer and Explanation is :
To estimate the distance traveled by the bicyclist using the Right Riemann Sum, we need to use the velocity data provided and break the time into three intervals of length 10 seconds each, as specified.
Here’s how we can approach this:
Step 1: Organize the Data
We are given the following time and velocity data:
| Time (sec) | Velocity (ft/sec) |
|---|---|
| 0 | 8 |
| 5 | 127 |
| 10 | 298 |
| 15 | 222 |
| 20 | 170 |
| 25 | 630 |
| 30 | 200 |
Step 2: Define Intervals
We will divide the 30 seconds into three intervals of 10 seconds each:
- Interval 1: From 0 to 10 seconds
- Interval 2: From 10 to 20 seconds
- Interval 3: From 20 to 30 seconds
Step 3: Apply the Right Riemann Sum
The Right Riemann Sum uses the velocity at the right end of each interval to estimate the distance.
Interval 1 (0 to 10 seconds):
- The right endpoint is at 10 seconds.
- Velocity at 10 seconds = 298 ft/sec.
- The length of the interval is 10 seconds.
So, the distance for this interval is:
[
\text{Distance} = \text{Velocity} \times \text{Time} = 298 \, \text{ft/sec} \times 10 \, \text{sec} = 2980 \, \text{ft}.
]
Interval 2 (10 to 20 seconds):
- The right endpoint is at 20 seconds.
- Velocity at 20 seconds = 170 ft/sec.
- The length of the interval is 10 seconds.
So, the distance for this interval is:
[
\text{Distance} = 170 \, \text{ft/sec} \times 10 \, \text{sec} = 1700 \, \text{ft}.
]
Interval 3 (20 to 30 seconds):
- The right endpoint is at 30 seconds.
- Velocity at 30 seconds = 200 ft/sec.
- The length of the interval is 10 seconds.
So, the distance for this interval is:
[
\text{Distance} = 200 \, \text{ft/sec} \times 10 \, \text{sec} = 2000 \, \text{ft}.
]
Step 4: Total Distance
Now, we can sum up the distances from each interval:
[
\text{Total Distance} = 2980 \, \text{ft} + 1700 \, \text{ft} + 2000 \, \text{ft} = 6680 \, \text{ft}.
]
Final Answer:
The estimated distance the bicyclist travels in 30 seconds, using the Right Riemann Sum, is 6680 feet.
Explanation:
The Right Riemann Sum is a numerical method for approximating the area under a curve, which in this case represents the distance traveled by the bicyclist. By dividing the total time into equal intervals (in this case, 10 seconds), and using the velocity at the right end of each interval, we can approximate the total distance. This method assumes that the velocity remains constant during each 10-second interval, even though the actual velocity might vary within the interval. By summing the results for each interval, we get the total estimated distance. This method is often used in situations where calculating an exact integral is not feasible.