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.