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Let's break down the given problem step by step to answer each part.

### Given Data
- **Time (seconds):** 0, 3, 5, 8, 13
- **Distance (feet):** 0, 225, 383, 623, 993
- **Speed (feet per second):** 75, 77, 80, 74, 72

### Part (a) - Hermite Polynomial for \( t = 10 \)
We are tasked with predicting the position and speed of the car at \( t = 10 \) seconds using a **Hermite polynomial**. A Hermite polynomial considers both position and velocity, which aligns with the information provided in the table (distance and speed).

To solve this, we will:
1. Use interpolation on both the distance and speed to predict values at \( t = 10 \).
2. Implement Hermite interpolation, which matches both function value (distance) and derivative value (speed).

### Part (b) - Speed exceeding 55 mi/h
We need to determine if the car exceeds a speed limit of **55 miles per hour**.
- Convert 55 miles per hour to feet per second:
  \[
  55 \text{ miles/hour} \times \frac{5280 \text{ feet}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}} = 80.67 \text{ feet/second}
  \]
- Analyze the speed values and use interpolation to find if and when the car exceeds this limit.

### Part (c) - Predicted Maximum Speed
We'll find the maximum predicted speed of the car, either from the given data or through interpolation.

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Let me begin with Part (a) by calculating the Hermite polynomial and predicting the car's position and speed at \( t = 10 \).### Part (a) Results:
Using Hermite interpolation:
- **Predicted position** of the car at \( t = 10 \) seconds: approximately **771.96 feet**.
- **Predicted speed** at \( t = 10 \) seconds: approximately **74.64 feet per second**.

### Moving on to Part (b):
We'll now determine whether the car ever exceeds the speed limit of **80.67 feet per second (55 miles per hour)** and, if so, find the time at which this occurs. Let me calculate that next.### Part (b) Results:
The car **exceeds the speed limit** of 55 miles per hour (80.67 feet per second) at approximately **5.21 seconds**, with a speed of around **80.75 feet per second**.

### Part (c) - Predicted Maximum Speed:
Next, let's find the predicted maximum speed of the car using interpolation. I'll calculate that now.### Part (c) Results:
The **predicted maximum speed** of the car is approximately **82.00 feet per second**, which occurs at around **5.99 seconds**.

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Would you like further details on any part of the solution or have any other questions?

Here are 5 related questions to explore further:
1. How is the Hermite polynomial different from other interpolation methods like Lagrange or cubic splines?
2. What would happen if we used only the speed values for interpolation instead of both speed and distance?
3. Can we extend this analysis to predict the total distance covered over a longer period, say \( t = 20 \) seconds?
4. How would using a different set of time points affect the accuracy of interpolation?
5. What role does the second derivative (acceleration) play in such Hermite interpolations?

**Tip:** When dealing with physical quantities like speed and position, ensuring the use of appropriate units in calculations is crucial for accurate results.

A car traveling along a straight road is clocked at a number of points. The data from the observations are given in the table, where the time is in seconds, the distance is in feet, and the speed is in feet per second. The tasks include predicting the position and speed of the car using Hermite polynomial, determining if the car exceeds 55 mph, and predicting the maximum speed.

Solve for the car's position and speed at 10 seconds using Hermite polynomial interpolation. Determine if the car exceeds 55 mph and find the predicted maximum speed. Learn about polynomial interpolation and speed analysis with step-by-step solutions.

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