The stopping distance of an automobile, on dry, level pavement, traveling at a speed v (in kilometers per hour) is the distance R (in meters) the car travels during the reaction time of the driver plus the distance B (in meters) the car travels after the brakes are applied (see figure). The table shows the results of the experiment. Speed, v	20	40	60	80	100
Reaction Time
Distance, R	8.8	17.2	25.5	33.8	42.2
Braking Time
Distance, B	2.8	9.5	20.7	36.3	56.4 Determine the polynomial giving the total stopping distance T. (Round numerical values to four decimal places.)

To determine the polynomial that represents the total stopping distance \( T(v) \), we can express it as the sum of two polynomials: the reaction time distance \( R(v) \) and the braking time distance \( B(v) \), both of which are given in the table.

### Total Stopping Distance Formula:
\[
T(v) = R(v) + B(v)
\]
Where:
- \( R(v) \) is the reaction time distance polynomial,
- \( B(v) \) is the braking time distance polynomial.

We are given the following data for speed \( v \) (in km/h), reaction time distance \( R(v) \), and braking time distance \( B(v) \):

| Speed \( v \) (km/h) | Reaction Time Distance \( R(v) \) (m) | Braking Time Distance \( B(v) \) (m) |
|--------------------|--------------------------------------|-------------------------------------|
| 20                 | 8.8                                  | 2.8                                 |
| 40                 | 17.2                                 | 9.5                                 |
| 60                 | 25.5                                 | 20.7                                |
| 80                 | 33.8                                 | 36.3                                |
| 100                | 42.2                                 | 56.4                                |

### Step 1: Fit a Polynomial to the Reaction Time Distance \( R(v) \)
We will fit a linear polynomial to the reaction time distance data, since reaction time generally increases linearly with speed. Assume the polynomial takes the form:
\[
R(v) = a_1 v + a_0
\]
We can use the given data points for \( R(v) \) to determine the coefficients \( a_1 \) and \( a_0 \) using regression techniques.

### Step 2: Fit a Polynomial to the Braking Time Distance \( B(v) \)
For the braking time distance, it is common for this to follow a quadratic relationship because braking distance tends to increase quadratically with speed. We will fit a quadratic polynomial:
\[
B(v) = b_2 v^2 + b_1 v + b_0
\]
We can use the given data points for \( B(v) \) to determine the coefficients \( b_2 \), \( b_1 \), and \( b_0 \).

I will calculate the coefficients for both polynomials now.The fitted polynomials for the reaction time distance \( R(v) \) and braking time distance \( B(v) \) are as follows:

### Reaction Time Distance Polynomial \( R(v) \):
\[
R(v) = 0.417 v + 0.48
\]

### Braking Time Distance Polynomial \( B(v) \):
\[
B(v) = 0.00557143 v^2 + 0.00142857 v + 0.54
\]

### Total Stopping Distance Polynomial \( T(v) \):
The total stopping distance is the sum of the reaction time distance and the braking time distance:
\[
T(v) = R(v) + B(v)
\]
\[
T(v) = (0.417 v + 0.48) + (0.00557143 v^2 + 0.00142857 v + 0.54)
\]
Simplifying this expression:
\[
T(v) = 0.00557143 v^2 + 0.41842857 v + 1.02
\]

Thus, the total stopping distance polynomial is:
\[
T(v) = 0.0056 v^2 + 0.4184 v + 1.02
\]

This is the polynomial that approximates the total stopping distance of the car based on its speed.

Would you like further clarification on this, or should I dive deeper into any part of the solution?

Here are some related questions you might want to explore:
1. How would the polynomial change for different road conditions (e.g., wet pavement)?
2. How can we interpret the coefficients of the reaction time distance polynomial?
3. What is the significance of the quadratic term in the braking distance polynomial?
4. Can we use this polynomial for speeds beyond 100 km/h?
5. How can we compare this stopping distance model with real-world measurements?

**Tip:** The braking distance typically increases quadratically with speed because kinetic energy increases with the square of velocity.

The stopping distance of an automobile, on dry, level pavement, traveling at a speed v (in kilometers per hour) is the distance R (in meters) the car travels during the reaction time of the driver plus the distance B (in meters) the car travels after the brakes are applied. The table shows the results of the experiment. Speed, v 20 40 60 80 100 Reaction Time Distance, R 8.8 17.2 25.5 33.8 42.2 Braking Time Distance, B 2.8 9.5 20.7 36.3 56.4 Determine the polynomial giving the total stopping distance T. (Round numerical values to four decimal places.)

Learn how to derive the polynomial equation representing the total stopping distance of a car based on speed. This problem uses data from an experiment, applying linear and quadratic equations to find a detailed solution. Suitable for students in Grades 10-12.

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