The figure below represents part of the performance data of a car owned by a proud physics student. (The horizontal axis is marked in increments of 1 s and the vertical axis is marked in increments of 10 m/s.)
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(a) Calculate the total distance traveled by computing the area under the red-brown graph line.
I've been trying to use the trapezoidal method to no avail on the area under the object, and I haven't learned how to integrate either in calculus, so can anyone help me out with how to actually go about solving this problem?
To calculate the total distance traveled by computing the area under the graph line, you can use numerical integration methods like the trapezoidal rule.
Here's a step-by-step approach to solving the problem using the trapezoidal rule:
1. Begin by dividing the graph into small rectangular sections. Each rectangle should have a width of 1s (as shown on the horizontal axis) and a height corresponding to the value on the vertical axis at that time.
2. Calculate the area of each rectangle by multiplying the width (1s) by the average height of the rectangle. This can be done by adding the heights at the two ends of the rectangle and then dividing by 2.
3. Sum up the areas of all the rectangles to find the total area under the graph. This will give you an estimate of the total distance traveled.
Let's apply this method to the given graph:
1. Divide the graph into small rectangles, as shown by the lines.
```
| | | | |
| | | | |
| | |_ _|_ _|
|___|___|___|___|
1s 2s 3s 4s
```
2. Calculate the area of each rectangle:
For the first rectangle (1s to 2s):
- The heights at the two ends of the rectangle are 0 m/s and 20 m/s.
- The average height is (0 + 20) / 2 = 10 m/s.
- The area of this rectangle is 1s * 10m/s = 10 m.
Repeat this calculation for each rectangle, and sum up all the areas.
3. Sum up the areas of all the rectangles to find the total area:
In this case, we have four rectangles with areas:
Rectangle 1: 10 m
Rectangle 2: 15 m
Rectangle 3: 10 m
Rectangle 4: 5 m
The total area under the graph, and hence the total distance traveled, is the sum of these areas:
Total distance traveled = 10 m + 15 m + 10 m + 5 m = 40 m.
Therefore, the total distance traveled by the car is 40 meters.
To calculate the total distance traveled by the car, you can approximate the area under the red-brown graph line using the trapezoidal rule. Here's how you can do it step by step:
Step 1: Divide the graph into smaller trapezoids. Approximate the shape of the graph line using straight lines between each data point. This will give you a series of trapezoids.
Step 2: Measure the base and height of each trapezoid. The base is the difference in time (horizontal axis) between two data points, and the height is the difference in velocity (vertical axis).
Step 3: Calculate the area of each trapezoid using the formula:
Area = (base1 + base2) * (height) / 2
where base1 and base2 are the lengths of the bases and height is the height of the trapezoid.
Step 4: Sum up the areas of all the trapezoids to find the total distance traveled.
Here's a step-by-step example using the given graph line:
Step 1: Divide the graph into smaller trapezoids. Let's assume the graph has five data points labeled A, B, C, D, and E.
Step 2: Measure the base and height of each trapezoid. For example:
- For trapezoid "ABCD", the base1 is 1 s (from point A to B), base2 is 1 s (from point C to D), and the height is 10 m/s.
- For trapezoid "BCDE", the base1 is 1 s (from point B to C), base2 is 2 s (from point D to E), and the height is 20 m/s.
Step 3: Calculate the area of each trapezoid:
- For trapezoid "ABCD":
Area_ABCD = (1 s + 1 s) * (10 m/s) / 2 = 2 s * 10 m/s / 2 = 10 m
- For trapezoid "BCDE":
Area_BCDE = (1 s + 2 s) * (20 m/s) / 2 = 3 s * 20 m/s / 2 = 30 m
Step 4: Sum up the areas of all the trapezoids to find the total distance traveled:
Total distance = Area_ABCD + Area_BCDE = 10 m + 30 m = 40 m
Therefore, the total distance traveled by the car, as represented by the area under the red-brown graph line, is 40 meters.