a radar gun was used to record the speed of a runner duriong the second of a race. Use simpsons rule to estimate the distance the runner covered in 5 seconds. The table has points (0,0), (0.5,3.15), (1,4.3), (1.5,5.45), (2,6.5), (2.5,7.45), (3,
8.95), (4,10.6), (4.5, 10.7), (5,10.7)

To estimate the distance the runner covered using Simpson's Rule, we treat the given points as measurements of speed at different times. The formula for Simpson's Rule, when applied to an odd number of intervals, is:

\[
\int_{a}^{b} f(x) dx \approx \frac{h}{3} \left[ y_0 + 4y_1 + 2y_2 + 4y_3 + \dots + 2y_{n-2} + 4y_{n-1} + y_n \right]
\]

Where:
- \( h = \frac{b-a}{n} \) is the width of each interval,
- \( y_0, y_1, \dots, y_n \) are the function values (speeds in this case),
- \( n \) is the number of subintervals, which must be even.

### Step-by-Step Solution:

1. **Data from the problem**:
   - Time (seconds): \( t = [0, 0.5, 1, 1.5, 2, 2.5, 3, 4, 4.5, 5] \)
   - Speed (meters per second): \( v = [0, 3.15, 4.3, 5.45, 6.5, 7.45, 8.95, 10.6, 10.7, 10.7] \)

2. **Identify \( h \)**:
   The time intervals are evenly spaced by \( h = 0.5 \) seconds.

3. **Apply Simpson's Rule formula**:
   Since we have 9 intervals (an even number, which is suitable for Simpson's rule), we can proceed as follows:

\[
\text{Distance} \approx \frac{h}{3} \left[ y_0 + 4y_1 + 2y_2 + 4y_3 + 2y_4 + 4y_5 + 2y_6 + 4y_7 + 2y_8 + y_9 \right]
\]

Substituting the values from the table:

\[
\text{Distance} \approx \frac{0.5}{3} \left[ 0 + 4(3.15) + 2(4.3) + 4(5.45) + 2(6.5) + 4(7.45) + 2(8.95) + 4(10.6) + 2(10.7) + 10.7 \right]
\]

4. **Calculate**:
   Simplifying inside the brackets:

\[
= 0.5 \times \frac{1}{3} \times \left[ 0 + 12.6 + 8.6 + 21.8 + 13 + 29.8 + 17.9 + 42.4 + 21.4 + 10.7 \right]
\]
\[
= 0.5 \times \frac{1}{3} \times \left[ 177.5 \right]
\]
\[
= 0.5 \times 59.167
\]
\[
= 29.5835 \text{ meters}
\]

### Conclusion:
The estimated distance the runner covered in 5 seconds, using Simpson's Rule, is approximately **29.58 meters**.

Would you like to explore any part of this in more detail or have any questions?

Here are 5 related questions to think about:
1. How does Simpson's rule compare with other numerical integration methods, like the trapezoidal rule?
2. What would happen if the time intervals were not evenly spaced?
3. Can Simpson's rule be applied to estimate the area under any curve?
4. How would errors accumulate if the given data were less accurate?
5. Can you apply Simpson's rule to other real-life problems, like estimating total fuel consumption from speed data?

**Tip**: Simpson’s rule works best when the data approximates a smooth curve; otherwise, more intervals might be necessary to improve accuracy.

A radar gun was used to record the speed of a runner during the second of a race. Use Simpson's rule to estimate the distance the runner covered in 5 seconds. The table has points (0,0), (0.5,3.15), (1,4.3), (1.5,5.45), (2,6.5), (2.5,7.45), (3,8.95), (4,10.6), (4.5,10.7), (5,10.7).

Use Simpson's rule to estimate the distance covered by a runner in 5 seconds based on speed data from a radar gun. This problem covers concepts like numerical integration and distance estimation, ideal for high school students (Grades 10-12).

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