Math Problem Statement
The graph shows a city's daily rate of use of electricity (in millions of kilowatts). Estimate the total daily usage of electricity by summing the area of rectangles. Use the left endpoints, then the right endpoints, then give the average of those results as the answer. Let the width of each rectangle be 2 hours. Midnight 4 A.M. 8 A.M. Noon 4 P.M. 8 P.M. Midnight
4 8 12 16 20 24 28
Millions of kilowatt-hours
A coordinate system has a horizontal axis labeled with hours from Midnight to Midnight, in increments of 4 hours and a vertical axis labeled Millions of kilowatt-hours from 0 to 24 in increments of 4. A smooth curve rises from right to left, passing through the points with the following y-coordinates, with the hours listed first: Midnight, 11.29; 4 A.M., 12.62; 8 A.M., 16.62; Noon, 23.29, to a maximum at 4 P.M, 29.15, and falls through the following points: 8 P.M., 20.49; Midnight, 13.48. All coordinates are approximate. Question content area bottom Part 1 Begin by finding the estimate of the total energy used on the day shown in the graph, by summing rectangles using left endpoints. The width of each interval is 2 hours.
Solution
To estimate the total daily electricity usage using the left endpoints, we can break the 24-hour period into 2-hour intervals and sum the areas of rectangles formed by these intervals.
Given Data
The points you provided for the electricity usage are:
- Midnight (0:00) - 11.29 million kWh
- 4 A.M. (4:00) - 12.62 million kWh
- 8 A.M. (8:00) - 16.62 million kWh
- Noon (12:00) - 23.29 million kWh
- 4 P.M. (16:00) - 29.15 million kWh
- 8 P.M. (20:00) - 20.49 million kWh
- Midnight (24:00) - 13.48 million kWh
Left Endpoint Calculation
Using the left endpoints, we take the height of the function at the beginning of each interval:
-
Midnight to 4 A.M. (0-4):
- Height = 11.29 million kWh
- Area = million kWh
-
4 A.M. to 8 A.M. (4-8):
- Height = 12.62 million kWh
- Area = million kWh
-
8 A.M. to Noon (8-12):
- Height = 16.62 million kWh
- Area = million kWh
-
Noon to 4 P.M. (12-16):
- Height = 23.29 million kWh
- Area = million kWh
-
4 P.M. to 8 P.M. (16-20):
- Height = 29.15 million kWh
- Area = million kWh
-
8 P.M. to Midnight (20-24):
- Height = 20.49 million kWh
- Area = million kWh
Total Area (Left Endpoint)
Now we can sum these areas:
Right Endpoint Calculation
Now, we calculate using the right endpoints. The heights will be taken from the end of each interval:
-
Midnight to 4 A.M. (0-4):
- Height = 12.62 million kWh
- Area = million kWh
-
4 A.M. to 8 A.M. (4-8):
- Height = 16.62 million kWh
- Area = million kWh
-
8 A.M. to Noon (8-12):
- Height = 23.29 million kWh
- Area = million kWh
-
Noon to 4 P.M. (12-16):
- Height = 29.15 million kWh
- Area = million kWh
-
4 P.M. to 8 P.M. (16-20):
- Height = 20.49 million kWh
- Area = million kWh
-
8 P.M. to Midnight (20-24):
- Height = 13.48 million kWh
- Area = million kWh
Total Area (Right Endpoint)
Now sum these areas:
Average of the Results
Finally, we find the average of the left and right endpoint estimates:
Summary
- Left Endpoint Estimate: 226.62 million kWh
- Right Endpoint Estimate: 231.60 million kWh
- Average Estimate: 229.11 million kWh
If you have any questions or need further clarification, feel free to ask! Here are some related questions you might consider:
- How would the total usage change if we used different width intervals?
- Can we analyze the data using more sophisticated numerical methods?
- How does electricity usage vary during different times of the day?
- What are the implications of this usage for energy policy?
- How can this data be visualized for better understanding?
Tip: For more accurate estimates, consider using more intervals or applying numerical integration techniques like trapezoidal or Simpson's rule.
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Math Problem Analysis
Mathematical Concepts
Calculus
Numerical Integration
Riemann Sums
Formulas
Area of Rectangle = height × width
Average = (Left Estimate + Right Estimate) / 2
Theorems
Mean Value Theorem for Integration
Suitable Grade Level
Grades 11-12
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