A surveyor measured the length of a piece of land at 100-ft intervals (x), as shown in the table. Use Simpson's Rule to estimate the area o the piece of land in square feet. Give the value of n on line 1. the Area on line 2. x Length (ft) 0 40 100 50 200 70 300 45 400 40
A surveyor measured the length of a piece of land at 100-ft intervals (x), as shown in the table. Use Simpson's Rule to estimate the area o the piece of land in square feet. Give the value of n on line 1. the Area on line 2. x Length (ft) 0 40 100 50 200 70 300 45 400 40
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Using Simpson's Rule to Estimate Land Area**
A surveyor measured the length of a piece of land at 100-ft intervals (\(x\)), as shown in the table below. Use Simpson's Rule to estimate the area of the piece of land in square feet. Provide the value of \(n\) on line 1 and the Area on line 2.
| \(x\) (ft) | Length (ft) |
|------------|-------------|
| 0 | 40 |
| 100 | 50 |
| 200 | 70 |
| 300 | 45 |
| 400 | 40 |
### Explanation
Simpson's Rule is a method for estimating the definite integral of a function. In this context, it's used to estimate the area under the curve representing the land's length measurements. Simpson's Rule states that:
\[
\int_{a}^{b} f(x) dx \approx \frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \ldots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)]
\]
For the given table:
- \(\Delta x\) is 100 feet (the interval between the measurements).
- \(n\) (the number of intervals) is 4.
Using Simpson's Rule,
\[
\text{Area} \approx \frac{100}{3} \times [40 + 4 \times 50 + 2 \times 70 + 4 \times 45 + 40]
\]
### Calculation
\[
\text{Area} \approx \frac{100}{3} \times [40 + 200 + 140 + 180 + 40]
\]
\[
\text{Area} \approx \frac{100}{3} \times 600 \]
\[
\text{Area} \approx 20000 \text{ square feet}
\]
Thus,
1. \( n = 4 \)
2. Area \(\approx 20000\) square feet](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a617c73-16c8-405c-b9c4-72dd4741d892%2Ff956d908-4225-4ac7-938e-3fbe126a9c34%2Fmpb2kbo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Using Simpson's Rule to Estimate Land Area**
A surveyor measured the length of a piece of land at 100-ft intervals (\(x\)), as shown in the table below. Use Simpson's Rule to estimate the area of the piece of land in square feet. Provide the value of \(n\) on line 1 and the Area on line 2.
| \(x\) (ft) | Length (ft) |
|------------|-------------|
| 0 | 40 |
| 100 | 50 |
| 200 | 70 |
| 300 | 45 |
| 400 | 40 |
### Explanation
Simpson's Rule is a method for estimating the definite integral of a function. In this context, it's used to estimate the area under the curve representing the land's length measurements. Simpson's Rule states that:
\[
\int_{a}^{b} f(x) dx \approx \frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \ldots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)]
\]
For the given table:
- \(\Delta x\) is 100 feet (the interval between the measurements).
- \(n\) (the number of intervals) is 4.
Using Simpson's Rule,
\[
\text{Area} \approx \frac{100}{3} \times [40 + 4 \times 50 + 2 \times 70 + 4 \times 45 + 40]
\]
### Calculation
\[
\text{Area} \approx \frac{100}{3} \times [40 + 200 + 140 + 180 + 40]
\]
\[
\text{Area} \approx \frac{100}{3} \times 600 \]
\[
\text{Area} \approx 20000 \text{ square feet}
\]
Thus,
1. \( n = 4 \)
2. Area \(\approx 20000\) square feet
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