Time (months) 0 1 2 3 4 5
Rate pollutants
are escaping 2 5 7 16 24 34
Use this data to sketch a graph
Draw rectangles on this graph (as in Figure 5.2) to help you underestimate the total pollutants that escaped during the first month. What underestimate did you obtain?
tons
Draw rectangles on this graph (as in Figure 5.2) to help you overestimate the total pollutants that escaped during the first month. What overestimate did you obtain?
tons
Draw rectangles on this graph (as in Figure 5.2) to help you underestimate the total pollutants that escaped during the whole time. What underestimate did you obtain?
tons
Draw rectangles on this graph (as in Figure 5.2) to help you overestimate the total pollutants that escaped during the whole time. What overestimate did you obtain?
tons
Hmm, drawing rectangles on a graph? Sounds like something an overenthusiastic art teacher would come up with. Anyway, let's give it a shot!
To underestimate the total pollutants that escaped during the first month, we can draw a rectangle that has a width equal to 1 (since it's the first month) and a height equal to the lowest rate of pollutants that escaped during that month, which is 2 tons. So, our rectangle has an area of 2 tons.
Now, to overestimate the total pollutants that escaped during the first month, we can draw a rectangle that covers the entire first month and has a height equal to the highest rate of pollutants that escaped during that month, which is 5 tons. Therefore, our rectangle has an area of 5 tons.
To underestimate the total pollutants that escaped during the whole time, we can sum up the areas of the rectangles for each month. So, we have rectangle 1 with an area of 2 tons, rectangle 2 with an area of 5 tons, rectangle 3 with an area of 7 tons, rectangle 4 with an area of 16 tons, rectangle 5 with an area of 24 tons, and rectangle 6 with an area of 34 tons. Adding all these up, we get an underestimate of 88 tons.
Now, let's overestimate the total pollutants that escaped during the whole time. We can draw a single rectangle that covers the entire time period and has a height equal to the highest rate of pollutants that escaped, which is 34 tons during the final month. So, our rectangle has an area of 34 tons.
There you have it! I hope my artistic interpretation was spot on. If not, blame my clown paint, not my artistic skills. Keep in mind that these are just rough estimates, so don't take them too seriously.
To sketch the graph, plot the time (months) on the x-axis and the rate of escaping pollutants on the y-axis.
Next, we will draw rectangles to estimate the total pollutants that escaped during specific time periods.
For underestimating the total pollutants that escaped during the first month, draw a rectangle with a base between time periods 0 and 1 and a height equal to the rate of escaping pollutants at time 0. This rectangle will have a value of 2 tons.
For overestimating the total pollutants that escaped during the first month, draw a rectangle with a base between time periods 0 and 1 and a height equal to the rate of escaping pollutants at time 1. This rectangle will have a value of 5 tons.
To underestimate the total pollutants that escaped during the whole time, you can draw rectangles with bases between each time period and heights equal to the rate of escaping pollutants at the starting time of each period. Add up the areas of these rectangles to find the total underestimate.
To overestimate the total pollutants that escaped during the whole time, you can draw rectangles with bases between each time period and heights equal to the rate of escaping pollutants at the ending time of each period. Again, add up the areas of these rectangles to find the total overestimate.
To sketch a graph using the given data, you need to plot the time (months) on the x-axis and the rate of pollutants escaping on the y-axis.
To underestimate the total pollutants that escaped during the first month, you need to draw a rectangle that lies entirely below the curve in the first month. The height of the rectangle should be the lowest rate of pollutants that escaped during the first month, and the width should be the time interval of the first month.
To calculate the underestimate, you can refer to the given data and find the lowest rate of pollutants that escaped during the first month. In this case, the lowest rate is 2 tons (at time 0). Multiply this rate by the time interval of the first month (1 month) to obtain the underestimate. So, the underestimate of the total pollutants that escaped during the first month is 2 tons.
To overestimate the total pollutants that escaped during the first month, you need to draw a rectangle that lies entirely above the curve in the first month. The height of the rectangle should be the highest rate of pollutants that escaped during the first month, and the width should be the time interval of the first month.
To calculate the overestimate, you can refer to the given data and find the highest rate of pollutants that escaped during the first month. In this case, the highest rate is 5 tons (at time 1). Multiply this rate by the time interval of the first month (1 month) to obtain the overestimate. So, the overestimate of the total pollutants that escaped during the first month is 5 tons.
To underestimate the total pollutants that escaped during the whole time, you need to draw rectangles below the curve for each time interval and sum up the areas of all the rectangles.
To calculate the underestimate, you can calculate the area of each rectangle using the width (time interval) and the corresponding lowest rate of pollutants for that interval. In this case, the rectangles will have widths of 1 month, 1 month, 1 month, 1 month, and 1 month, and the corresponding lowest rates of pollutants are 2 tons, 5 tons, 7 tons, 16 tons, and 24 tons, respectively. Add up the areas of all the rectangles to obtain the underestimate. So, the underestimate of the total pollutants that escaped during the whole time is 2 + 5 + 7 + 16 + 24 = 54 tons.
To overestimate the total pollutants that escaped during the whole time, you need to draw rectangles above the curve for each time interval and sum up the areas of all the rectangles.
To calculate the overestimate, you can calculate the area of each rectangle using the width (time interval) and the corresponding highest rate of pollutants for that interval. In this case, the rectangles will have widths of 1 month, 1 month, 1 month, 1 month, and 1 month, and the corresponding highest rates of pollutants are 5 tons, 7 tons, 16 tons, 24 tons, and 34 tons, respectively. Add up the areas of all the rectangles to obtain the overestimate. So, the overestimate of the total pollutants that escaped during the whole time is 5 + 7 + 16 + 24 + 34 = 86 tons.