Chapter 6.1, Problem 29E

(a)

To determine

The value of lower estimate.

Answer to Problem 29E

The upper estimate is 60.9 tons and lower estimate is 46.8 tons.

Explanation of Solution

Given information:

  

Formula used:

The right-hand endpoint rectangular approximation method is used.

Calculation:

RRAM means right-hand endpoint rectangular approximation method. The name suggests the choice we made when determining the heights of the approximating rectangles, hence evaluated the function at the right-hand endpoint of each subinterval.

Since the release rate of pollutants is increasing, an upper estimate is given by using the data for the end of each month (right rectangles), assuming that new scrubbers were installed before the beginning of January.

  $\begin{array}{l}\text{upper}\text{estimate}=30\left(0.20+0.25+0.27+0.34+0.45+0.52\right)\\ \text{upper}\text{estimate}=6+7.5+8.1+10.2+13.5+15.6\end{array}$

These values tonnage of pollutants at the end of month,

  $\text{upper estimate}=60.9\text{tons of pollutants}$

LRAM means left-hand endpoint rectangular approximation method. The name suggests the choice we made when determining the heights of the approximating rectangles, hence evaluated the function at the left-hand endpoint of each subinterval.

A lower estimate is given by using the data for the end of the previous month. The no data for the beginning of January, but those pollutants were released at the new-scrubber rate of 0.5 ton/day. 80 we may use this value,

  $\begin{array}{l}\text{lower}\text{estimate}=30\left(0.05+0.20+0.25+0.27+0.34+0.54\right)\\ =15+6+7.5+8.1+10.2+13.5\\ \text{lower}\text{estimate}=46.8\text{tons of pollutants}\end{array}$

Conclusion:

The upper estimate is 60.9 tons and lower estimate is 46.8 tons.

(b)

To determine

The month that 125 tons will have been released into the atmosphere.

Answer to Problem 29E

A total of 125 tons will have been released into the atmosphere by the end of October.

Explanation of Solution

Given information:

  

Formula used:

The left-hand endpoint rectangular approximation method is used.

Calculation:

Using LRAM, the amount of pollutants released by the end of October is

  $\begin{array}{l}\text{amount of pollutants}=30\left(0.05+0.20+0.25+0.27+0.34+0.45+0.52+0.63+0.70+0.81\right)\\ =16+6+7.5+8.1+10.2+13.5+15.6+18.9+21.0+24.3\\ \text{amount of pollutants}=126.6\text{tons}\end{array}$

So, a total of 125 tons will have been released into the atmosphere by the end of October.

Conclusion:

A total of 125 tons will have been released into the atmosphere by the end of October.