Time (months)| 0 | 1 | 2 | 3 | 4 | 5 |
Rate pollutants are escaping (tons/month)| 4 | 6 | 10 | 13 | 17 | 32 |
1) Draw rectangles on this graph to help you underestimate the total pollutants that escaped during the whole time. What underestimate did you obtain?
2) Draw rectangles on this graph to help you overestimate the total pollutants that escaped during the whole time. What overestimate did you obtain?
since the slope of the graph is generally increasing, left-endpoints will underestimate
So, for the first 5 points, draw a rectangle to the right at those points.
4+6+10+13+17 = ____
and of course, for the last 5 points, draw rectangles to the left to overestimate
6+10+13+17+32 = ____
also, I suspect that 32 should really be 22, since it fits better with the other values.
To estimate the total pollutants that escaped during the whole time, we can draw rectangles on the graph using the given data.
1) Underestimate:
To underestimate the total pollutants that escaped, we can draw rectangles that are entirely below the curve. In this case, we will connect the points with straight lines to create rectangles.
We start by drawing a rectangle for each time interval (0-1, 1-2, 2-3, 3-4, 4-5) using the height of the rectangle as the minimum value between the two points. The width of each rectangle represents the time interval.
For example:
- Rectangle 1: Height = minimum(4, 6) = 4, Width = 1
- Rectangle 2: Height = minimum(6, 10) = 6, Width = 1
- Rectangle 3: Height = minimum(10, 13) = 10, Width = 1
- Rectangle 4: Height = minimum(13, 17) = 13, Width = 1
- Rectangle 5: Height = minimum(17, 32) = 17, Width = 1
Next, we calculate the area of each rectangle:
- Area1 = 4 * 1 = 4
- Area2 = 6 * 1 = 6
- Area3 = 10 * 1 = 10
- Area4 = 13 * 1 = 13
- Area5 = 17 * 1 = 17
Finally, we sum up the areas of all the rectangles to obtain the total underestimate:
Total Underestimate = Area1 + Area2 + Area3 + Area4 + Area5 = 4 + 6 + 10 + 13 + 17 = 50 tons
Therefore, the total underestimate of pollutants that escaped during the whole time is 50 tons.
2) Overestimate:
To overestimate the total pollutants that escaped, we can draw rectangles that are entirely above the curve. In this case, we will connect the points with straight lines again to create rectangles.
Using the same technique as before, we draw rectangles for each time interval:
- Rectangle 1: Height = maximum(4, 6) = 6, Width = 1
- Rectangle 2: Height = maximum(6, 10) = 10, Width = 1
- Rectangle 3: Height = maximum(10, 13) = 13, Width = 1
- Rectangle 4: Height = maximum(13, 17) = 17, Width = 1
- Rectangle 5: Height = maximum(17, 32) = 32, Width = 1
Calculating the area of each rectangle:
- Area1 = 6 * 1 = 6
- Area2 = 10 * 1 = 10
- Area3 = 13 * 1 = 13
- Area4 = 17 * 1 = 17
- Area5 = 32 * 1 = 32
Summing up the areas of all the rectangles to obtain the total overestimate:
Total Overestimate = Area1 + Area2 + Area3 + Area4 + Area5 = 6 + 10 + 13 + 17 + 32 = 78 tons
Therefore, the total overestimate of pollutants that escaped during the whole time is 78 tons.
To draw the rectangles on the graph, we need to consider the width of each rectangle as the horizontal difference between two consecutive time points. The height of each rectangle represents the rate of pollutants escaping at that time, multiplied by the width.
1) Underestimating the total pollutants that escaped during the whole time:
To underestimate, we will use rectangles that are shorter than the actual rate of pollutants escaping at each time point. We can choose the lower value of each rate and multiply it by the width.
For example, let's consider the first rectangle between time 0 and time 1. The width is 1 month, and the minimum rate is 4 tons/month. So, the area of this rectangle is 1 * 4 = 4 tons. Similarly, we can calculate the areas of the other rectangles and add them up.
The underestimate is the sum of all the areas of the rectangles:
Underestimate = 1 * 4 + 1 * 6 + 1 * 10 + 1 * 13 + 1 * 17 + 1 * 32 = 82 tons
2) Overestimating the total pollutants that escaped during the whole time:
To overestimate, we will use rectangles that are taller than the actual rate of pollutants escaping at each time point. We can choose the maximum value of each rate and multiply it by the width.
Using the same process, we can calculate the areas of the rectangles using the maximum values for each rate:
Overestimate = 1 * 6 + 1 * 10 + 1 * 13 + 1 * 17 + 1 * 32 = 78 tons
Therefore, the underestimate is 82 tons, and the overestimate is 78 tons.