Calculus Page
Welcome to the Calculus Page. This page contains problems designed for high school students taking a first year Calculus course.
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| What's the area? |
A certain lake has an irregular shape. In order to estimate the area, the North-South distance across the lake is measured every 100 feet down an East-West line. The results are shown in the table below. Use an appropriate technique to estimate the number of acres that the lake covers. Give a complete explanation of how you computed the area.
| East-West position (ft) |
Distance across lake (ft) |
| 0 |
900 |
100 |
1500 |
200 |
1800 |
300 |
2000 |
400 |
2500 |
500 |
2700 |
600 |
2700 |
700 |
2700 |
800 |
2700 |
900 |
2700 |
1000 |
2700 |
1100 |
2700 |
1200 |
2700 |
1300 |
2700 |
1400 |
2600 |
1500 |
2600 |
1600 |
2600 |
1700 |
2600 |
1800 |
2600 |
1900 |
2600 |
2000 |
2500 |
2100 |
2500 |
2200 |
2500 |
2300 |
2500 |
2400 |
2500 |
2500 |
2500 |
2600 |
2400 |
2700 |
2300 |
2800 |
2300 |
2900 |
1800 |
3000 |
1600 |
3100 |
1500 |
3200 |
1400 |
3300 |
1400 |
3400 |
1300 |
3500 |
1300 |
3600 |
1100 |
3700 |
800 |
3800 |
0 |
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Problem Moderated by: MrT |
| Solution |
There are multiple ways one might attempt the problem. We could do a Riemann Sum, use the Trapezoidal Rule, or use Simpson's Rule. Since a typical lake would be curved, it suggests that Simpson's Rule might be the most appropriate method, although all three methods end up giving basically the same result.
Using Simpson's Rule, one would add the first distance (900), plus four times the second distance (1500), plus twice the third (1800), plus four times the fourth (2000), and so forth, alternating back and forth between two times and four times. We end with four times the next to last number (800) plus the last number (0). The entire sum is then multiplied by (1/3) and by 100 (the width of the interval), to arrive at an estimate of 8,250,000 square feet. Dividing by 43,560 square feet per acre gives a result of 189.39 acres, or 189 acres.
Using the Trapezoidal Rule gives an almost identical answer of 189.05 acres. Using rectangles, and doing a Riemann sum also gives a comparable answer. |
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