To estimate the distance the runner covered during the first 5 seconds of the race using Simpson's Rule, we first need to apply the formula for Simpson's Rule, which is:

$\text{Distance} = \frac{\Delta t}{3} \left[ v(t_0) + 4 \sum_{i=1}^{n-1} v(t_i) + v(t_n) \right]$

Where:

• $v(t_0)$ is the velocity at the first time value,
• $v(t_n)$ is the velocity at the last time value,
• $v(t_i)$ are the velocities at the intermediate time points,
• $\Delta t$ is the time interval, which is constant for Simpson's Rule.

Data provided:

t (s)	v (m/s)
0	0
0.5	4.67
1.0	7.34
1.5	8.82
2.0	9.73
2.5	10.22
3.0	10.51
3.5	10.69
4.0	10.72
4.5	10.81
5.0	10.81

Step-by-step process:

1. Choose time intervals ($\Delta t$): The time intervals are equally spaced with a gap of 0.5 seconds.

2. Apply Simpson's Rule: We will use the velocities provided at each $t$-value for our calculations. Since the table provides data at 11 equally spaced points from $t = 0$ to $t = 5$, we apply Simpson's Rule with these values.

$\text{Distance} = \frac{0.5}{3} \left[ v(0) + 4 \left( v(0.5) + v(1.0) + v(1.5) + v(2.0) + v(2.5) + v(3.0) + v(3.5) + v(4.0) + v(4.5) \right) + v(5.0) \right]$

Plugging in the values:

$\text{Distance} = \frac{0.5}{3} \left[ 0 + 4 \left( 4.67 + 7.34 + 8.82 + 9.73 + 10.22 + 10.51 + 10.69 + 10.72 + 10.81 \right) + 10.81 \right]$

3. Calculate the sum:

$\text{Sum of velocities} = 4.67 + 7.34 + 8.82 + 9.73 + 10.22 + 10.51 + 10.69 + 10.72 + 10.81 = 82.81$

4. Final computation:

$\text{Distance} = \frac{0.5}{3} \left[ 0 + 4(82.81) + 10.81 \right]$ $= \frac{0.5}{3} \left[ 0 + 331.24 + 10.81 \right]$ $= \frac{0.5}{3} \times 342.05$ $= \frac{171.025}{3} = 57.01 \text{ meters}$

Conclusion:

The estimated distance the runner covered during the first 5 seconds of the race is approximately 57.01 meters.

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Would you like further clarification on Simpson's Rule or any other details? Here are some related questions for you to explore:

1. How does Simpson's Rule compare to other numerical integration methods like the Trapezoidal Rule?
2. What happens to the accuracy of Simpson's Rule as the number of intervals increases?
3. How would you estimate the distance if the time intervals were unequal?
4. What are some real-world applications of numerical integration methods like Simpson's Rule?
5. How can you derive Simpson’s Rule from Newton-Cotes formulas?

Tip: When using Simpson's Rule, ensure that the number of intervals is even to get the correct results!