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The problem involves using **Simpson's Rule** to estimate the distance covered by a runner based on velocity data over a 5-second interval. Here’s a breakdown of each step to find the answer:

### Step 1
- We need to approximate the integral \( \int_0^5 v(t) \, dt \) to find the distance covered.
- **Simpson's Rule** will be used with \( \Delta t = \frac{1}{2} = 0.5 \) and \( n = 10 \) subintervals.

### Step 2
The formula for Simpson's Rule is:
\[
\int_a^b f(x) \, dx \approx S_n = \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n) \right]
\]
where:
- \( \Delta x = 0.5 \)
- \( f(x) = v(t) \), which we obtain from the velocity values in the table at each time point.

### Step 3
Plugging in the values from the table for \( v(t) \):
\[
\text{distance} = \int_0^5 v(t) \, dt \approx \frac{1}{6} \left( 0 + 4(4.64) + 2(7.33) + 4(8.85) + \ldots + 10.88 \right)
\]

Following through with this computation will yield the approximate distance covered over the 5-second interval. Would you like a detailed solution with step-by-step calculations?

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### Related Questions
1. How does Simpson’s Rule improve accuracy over other numerical integration methods like the Trapezoidal Rule?
2. What are the conditions for using Simpson’s Rule, and why are they important?
3. How would the result change if we used a different number of subintervals?
4. Can Simpson's Rule be applied to non-uniform data spacing, and if not, why?
5. How would we apply a different numerical method, like the Midpoint Rule, to approximate this integral?

### Tip
For Simpson’s Rule, ensure the number of intervals \( n \) is even to apply the formula correctly.

A radar gun was used to record the speed of a runner during the first 5 seconds of a race. Using the provided data table, use Simpson's Rule to estimate the distance the runner covered during those 5 seconds, rounded to three decimal places.

Learn to estimate the distance covered by a runner over 5 seconds using Simpson's Rule based on velocity data. This problem includes numerical integration and approximation techniques, suitable for high school students in Grades 11-12.

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