The accompanying table shows​ time-to-speed data for a particular car accelerating from rest to 120120 mph. How far  had the car traveled by the time it reached this​ speed? (Use trapezoids to estimate the area under the velocity curve. Notice that intervals vary in​ length.)
Speed change
Time​ (sec)
nbsp Zero to nbsp    Zero to 
2020 mphmph
2.22.2
3030 mphmph
3.33.3
4040 mphmph
4.74.7
5050 mphmph
6.26.2
6060 mphmph
8.28.2
7070 mphmph
10.710.7
8080 mphmph
13.313.3
9090 mphmph
16.716.7
100100 mphmph
21.421.4
110110 mphmph
27.127.1
120120 mphmph
38.138.1
.
.
.
Question content area right
Part 1
The car will travel enter your response here ft by the time it reaches this speed.
​(Round the final answer to three decimal places as needed. Round all intermediate values to five decimal places as​ needed.)

To estimate the distance traveled by the car, we'll use the trapezoidal rule to approximate the area under the velocity-time graph, which gives us the displacement (or distance traveled).

The basic idea is to treat the speed-time data as forming trapezoids between the given points. The area of each trapezoid will give an estimate of the distance traveled over each time interval. The total distance traveled is the sum of all trapezoid areas.

### Formula for Trapezoidal Rule:
For each interval, the area of a trapezoid is given by:

\[
\text{Area} = \frac{1}{2} \times ( \text{Speed at first point} + \text{Speed at second point}) \times \text{Time difference}
\]

Given that we have the following time and speed dataTo estimate the distance traveled by the car, we'll use the trapezoidal rule to approximate the area under the velocity-time graph, which gives us the displacement (or distance traveled).

The basic idea is to treat the speed-time data as forming trapezoids between the given points. The area of each trapezoid will give an estimate of the distance traveled over each time interval. The total distance traveled is the sum of all trapezoid areas.

### Formula for Trapezoidal Rule:
For each interval, the area of a trapezoid is given by:

\[
\text{Area} = \frac{1}{2} \times ( \text{Speed at first point} + \text{Speed at second point}) \times \text{Time difference}
\]

Given that we have the following time and speed 

| Speed (mph) | Time (sec) |
|-------------|------------|
| 0 mph       | 0 sec      |
| 20 mph      | 2.2 sec    |
| 30 mph      | 3.3 sec    |
| 40 mph      | 4.7 sec    |
| 50 mph      | 6.2 sec    |
| 60 mph      | 8.2 sec    |
| 70 mph      | 10.7 sec   |
| 80 mph      | 13.3 sec   |
| 90 mph      | 16.7 sec   |
| 100 mph     | 21.4 sec   |
| 110 mph     | 27.1 sec   |
| 120 mph     | 38.1 sec   |

We'll apply the trapezoidal rule to each segment.

#### Step-by-Step Calculation:

1. **Convert speeds to ft/sec**: 
   1 mph ≈ 1.46667 ft/sec

2. **Calculate areas for each interval using the trapezoidal formula**.

Let me compute this.The car traveled approximately 4741.011 feet by the time it reached 120 mph.

Would you like further details or have any questions? Here are 5 related questions:

1. How does the trapezoidal rule approximate the area under a curve?
2. Why do we need to convert speeds from mph to ft/sec when calculating distance?
3. What other methods can be used to estimate areas under curves besides the trapezoidal rule?
4. How would the estimated distance change if we had more precise data points?
5. How does acceleration impact the shape of the velocity-time curve?

**Tip:** The trapezoidal rule works well for approximating areas, but more precise methods like Simpson's rule might yield better accuracy with a higher number of intervals.

The accompanying table shows time-to-speed data for a particular car accelerating from rest to 120 mph. How far had the car traveled by the time it reached this speed? (Use trapezoids to estimate the area under the velocity curve. Notice that intervals vary in length.)

Learn how to use the trapezoidal rule to estimate the distance a car travels while accelerating from rest to 120 mph. This problem involves numerical integration, converting speeds from mph to ft/sec, and applying the trapezoidal formula to a velocity-time graph. Suitable for students in Grades 10-12.

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