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The problem involves a car traveling along a straight road, with data for time, distance, and speed provided in the table. The task is to use **Hermite polynomials** to predict the position and speed at a certain time, determine if the speed limit is exceeded, and find the maximum speed of the car. Let's break this down.

**Given dataThe problem involves a car traveling along a straight road, with data for time, distance, and speed provided in the table. The task is to use **Hermite polynomials** to predict the position and speed at a certain time, determine if the speed limit is exceeded, and find the maximum speed of the car. Let's break this down.

**Given **

- **Time (t)** in seconds: \( 0, 3, 5, 8, 13 \)
- **Distance (d)** in feet: \( 0, 225, 383, 623, 993 \)
- **Speed (v)** in feet per second: \( 75, 77, 80, 74, 72 \)

### Questions:
**a. Use a Hermite polynomial to predict the position of the car and its speed when \( t = 10 \, \text{s} \).**

Hermite interpolation uses both position (distance) and derivative (speed) data. Given the values, we can construct the polynomial and evaluate it at \( t = 10 \, \text{s} \). This will involve solving a Hermite interpolation problem with both position and speed data at each time point.

**b. Use the derivative of the Hermite polynomial to determine whether the car ever exceeds a 55-mi/h speed limit on the road. If so, what is the first time the car exceeds this speed?**

The speed limit of 55 mi/h is equivalent to \( 55 \times \frac{5280}{3600} = 80.67 \, \text{feet per second} \). We will differentiate the Hermite polynomial and check if the speed exceeds this value, then find the first time it occurs.

**c. What is the predicted maximum speed for the car?**

By finding the critical points from the derivative of the Hermite polynomial, we can determine the maximum speed.

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I will begin by constructing the Hermite polynomial for the position and speed using the given data points and calculate the necessary predictions. Let's start with part (a). I will compute this now.For part **(a)**:

- The predicted position of the car at \( t = 10 \, \text{seconds} \) is approximately **771.96 feet**.
- The predicted speed at \( t = 10 \, \text{seconds} \) is approximately **74.64 feet per second**.

I will now calculate part **(b)**, determining if the car ever exceeds the speed limit of 55 mi/h (which is 80.67 feet per second) and when it happens.For part **(b)**:
- The car first exceeds the speed limit of 80.67 feet per second at approximately **5.19 seconds**.

For part **(c)**:
- The predicted maximum speed for the car is approximately **82.00 feet per second**.

Would you like more details on any part of the solution or have any further questions? Here are some related questions you might explore:

1. How are Hermite polynomials different from Lagrange polynomials?
2. What is the significance of using both position and speed in Hermite interpolation?
3. How is the derivative of a Hermite polynomial computed?
4. How would this interpolation change if more data points were added?
5. What are other applications of Hermite interpolation in physics or engineering?

**Tip**: Always double-check units, especially when dealing with speed limits in different systems (e.g., mi/h vs. feet per second).

A car traveling along a straight road is clocked at a number of points. The data from the observations are given in the following table, where the time is in seconds, the distance is in feet, and the speed is in feet per second. 
Time (t): 0, 3, 5, 8, 13 
Distance (d): 0, 225, 383, 623, 993 
Speed (v): 75, 77, 80, 74, 72 
a. Use a Hermite polynomial to predict the position of the car and its speed when t = 10 s. 
b. Use the derivative of the Hermite polynomial to determine whether the car ever exceeds a 55-mi/h speed limit on the road. If so, what is the first time the car exceeds this speed? 
c. What is the predicted maximum speed for the car?

This problem explores using Hermite polynomials to predict a car's position and speed at various times, check if the speed exceeds a limit, and calculate the maximum speed using given time, distance, and speed data. Learn the detailed mathematical approach to Hermite interpolation, differentiation, and finding critical points.

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