A Radar Gun Was Used To Record The Speed Of A Runner During The First 5 Seconds Of A Race (see The Table Below). Use Simpson's Rule To Estimate The Distance The Runner Covered During Those 5 Seconds. Round Your Answer To Three Decimal
Introduction
In this article, we will use Simpson's Rule to estimate the distance a runner covered during the first 5 seconds of a race. The speed of the runner was recorded using a radar gun, and we will use this data to calculate the distance traveled. Simpson's Rule is a numerical method used to approximate the value of a definite integral, which in this case will give us the distance covered by the runner.
The Data
| Time (s) | Speed (m/s) |
|---|---|
| 0 | 0 |
| 0.5 | 2.5 |
| 1 | 4.5 |
| 1.5 | 6.5 |
| 2 | 8.5 |
| 2.5 | 10.5 |
| 3 | 12.5 |
| 3.5 | 14.5 |
| 4 | 16.5 |
| 4.5 | 18.5 |
| 5 | 20.5 |
Simpson's Rule
Simpson's Rule is a method for approximating the value of a definite integral. It is based on the idea of approximating the area under a curve by dividing it into small parabolic segments. The rule states that if we have a function f(x) and we want to approximate the value of the definite integral from a to b, we can use the following formula:
∫[a,b] f(x) dx ≈ (h/3) * [f(x0) + 4f(x1) + 2f(x2) + ... + 4f(xn-1) + f(xn)]
where h is the width of each subinterval, and x0, x1, ..., xn are the points at which we evaluate the function.
Applying Simpson's Rule to the Data
In this case, we want to approximate the distance covered by the runner during the first 5 seconds of the race. We can use Simpson's Rule to do this by approximating the definite integral of the speed function over the time interval from 0 to 5 seconds.
First, we need to divide the time interval into small subintervals. Let's use 10 subintervals, each of width 0.5 seconds. This means that we will evaluate the speed function at 11 points: 0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, and 5.
Next, we need to evaluate the speed function at each of these points. We can do this by looking at the table above.
| Time (s) | Speed (m/s) |
|---|---|
| 0 | 0 |
| 0.5 | 2.5 |
| 1 | 4.5 |
| 1.5 | 6.5 |
| 2 | 8.5 |
| 2.5 | 10.5 |
| 3 | 12.5 |
| 3.5 | 14.5 |
| 4 | 16.5 |
| 4.5 | 18.5 |
| 5 | 20.5 |
Now, we can apply Simpson's Rule to approximate the definite integral of the speed function over the time interval from 0 to 5 seconds.
∫[0,5] v(t) dt ≈ (0.5/3) * [0 + 4(2.5) + 2(4.5) + 4(6.5) + 2(8.5) + 4(10.5) + 2(12.5) + 4(14.5) + 2(16.5) + 4(18.5) + 20.5]
Calculating the Definite Integral
Now, we can calculate the definite integral using the formula above.
∫[0,5] v(t) dt ≈ (0.5/3) * [0 + 10 + 9 + 26 + 17 + 42 + 25 + 58 + 33 + 74 + 20.5]
∫[0,5] v(t) dt ≈ (0.5/3) * 304.5
∫[0,5] v(t) dt ≈ 50.75
Conclusion
In this article, we used Simpson's Rule to estimate the distance a runner covered during the first 5 seconds of a race. We recorded the speed of the runner using a radar gun and used this data to calculate the distance traveled. Simpson's Rule is a numerical method used to approximate the value of a definite integral, and in this case, it gave us an estimate of the distance covered by the runner.
Final Answer
The final answer is: 50.750
Q&A: Simpson's Rule and the Runner's Distance
Q: What is Simpson's Rule and how is it used?
A: Simpson's Rule is a numerical method used to approximate the value of a definite integral. It is based on the idea of approximating the area under a curve by dividing it into small parabolic segments. In this case, we used Simpson's Rule to estimate the distance a runner covered during the first 5 seconds of a race.
Q: How do you apply Simpson's Rule to a set of data?
A: To apply Simpson's Rule, you need to divide the time interval into small subintervals, evaluate the function at each of these points, and then use the formula to approximate the definite integral.
Q: What is the formula for Simpson's Rule?
A: The formula for Simpson's Rule is:
∫[a,b] f(x) dx ≈ (h/3) * [f(x0) + 4f(x1) + 2f(x2) + ... + 4f(xn-1) + f(xn)]
where h is the width of each subinterval, and x0, x1, ..., xn are the points at which we evaluate the function.
Q: How many subintervals did you use in this example?
A: We used 10 subintervals, each of width 0.5 seconds.
Q: What was the final answer for the distance covered by the runner?
A: The final answer was approximately 50.750 meters.
Q: What are some common applications of Simpson's Rule?
A: Simpson's Rule is commonly used in a variety of fields, including physics, engineering, and economics. It is often used to approximate the value of definite integrals, which can be used to solve problems involving area, volume, and other quantities.
Q: What are some advantages of using Simpson's Rule?
A: Simpson's Rule is a powerful tool for approximating definite integrals. It is often more accurate than other numerical methods, such as the trapezoidal rule, and it can be used to approximate a wide range of functions.
Q: What are some disadvantages of using Simpson's Rule?
A: One disadvantage of Simpson's Rule is that it requires a large number of function evaluations, which can be time-consuming and computationally intensive. Additionally, Simpson's Rule may not be as accurate as other methods, such as the Monte Carlo method, for certain types of functions.
Q: Can Simpson's Rule be used to approximate the value of a definite integral with an infinite number of subintervals?
A: No, Simpson's Rule is typically used to approximate the value of a definite integral with a finite number of subintervals. However, it is possible to use Simpson's Rule in conjunction with other numerical methods, such as the Monte Carlo method, to approximate the value of a definite integral with an infinite number of subintervals.
Q: How can Simpson's Rule be used in real-world applications?
A: Simpson's Rule can be used in a variety of real-world applications, including:
- Calculating the area under a curve
- Calculating the volume of a solid
- Calculating the work done by a force
- Calculating the energy of a system
- Calculating the distance traveled by an object
Q: What are some common mistakes to avoid when using Simpson's Rule?
A: Some common mistakes to avoid when using Simpson's Rule include:
- Not using enough subintervals
- Not evaluating the function at the correct points
- Not using the correct formula
- Not checking the accuracy of the result
Q: How can Simpson's Rule be used to solve problems involving area and volume?
A: Simpson's Rule can be used to approximate the area under a curve and the volume of a solid by dividing the area or volume into small subintervals and evaluating the function at each of these points.
Q: How can Simpson's Rule be used to solve problems involving work and energy?
A: Simpson's Rule can be used to approximate the work done by a force and the energy of a system by dividing the work or energy into small subintervals and evaluating the function at each of these points.
Q: What are some common applications of Simpson's Rule in physics and engineering?
A: Simpson's Rule is commonly used in physics and engineering to approximate the value of definite integrals, which can be used to solve problems involving area, volume, work, and energy.
Q: What are some common applications of Simpson's Rule in economics?
A: Simpson's Rule is commonly used in economics to approximate the value of definite integrals, which can be used to solve problems involving the behavior of economic systems.
Q: What are some common applications of Simpson's Rule in computer science?
A: Simpson's Rule is commonly used in computer science to approximate the value of definite integrals, which can be used to solve problems involving the behavior of algorithms and data structures.