Hydrologic scientists are trying to increase the safety of the drinking water for a local community.

Question

The scientists add a filtering system to all wells in the community and chart the contaminants in the water over time. The following graph shows the contaminants, in parts per billion (ppb), that were found in the water for a number of days after the filtering systems were added.

 

Part A: How many days did it take the filtering system to significantly decrease the contaminants?

Part B: Which option correctly describes the domain of the function? 

Select one answer that is completely correct for Part A and one answer that is completely correct for Part B.

B: The domain of the function is (4,∞)

, which describes the contaminant parts per billion.

A: The filtering system never made a significant effect, because the contaminant parts per billion increases over time.

A: The filtering system took approximately 4

 days to take effect, indicated by the x

-intercept.

B: The domain of the function is (−∞,∞)

, which describes the contaminant parts per billion.

A: The filtering system took approximately 4

 days to take effect, indicated by the vertical asymptote.

B: The domain of the function is (−∞,∞)

, which describes the time elapsed in days since the beginning of the study.

A: The filtering system took approximately 1200

 days to take effect, indicated by the vertical asymptote.

B: The domain of the function is (4,∞)

, which describes the time elapsed in days since the beginning of the study. 

Answer 1

A lot of times in test questions it is not about choosing the correct answer, but the best answer.

Part A:

I would choose "The filtering system took approximately 4 days to take effect, indicated by the vertical asymptote." My reasoning is as follows: The shape of the graph resembles f(x) = e^(-x) but with a shifted vertical asymptote. We can observe that the slope of the tangent line to the curve begins very steeply negative and then slowly becomes less negative and eventually almost zero to indicate no significant change after 20 days. To be brief, the significant decrease is wherever the slope of the tangent line to the curve is steepest AND negative. With this reasoning, choice "The filtering system took approximately 4 days to take effect, indicated by the vertical asymptote." seems the best.

Part B:

First let's define our terms. Domain is the set of all x values for a given function or curve. Now based on our answer for Part A we know there is a vertical asymptote around x = 4 days which means that there are no x-values to the left of OR including 4. Furthermore, we can observe that the function continues to the right after 20 days with no vertical asymptote indicated so we can assume it continues to ∞. Therefore, before even looking at the choices I would answer that the domain of the function is (4 , ∞) where 4 is NOT inclusive. Looking at the choices given, I would choose "The domain of the function is (4,∞), which describes the time elapsed in days since the beginning of the study."; these answers also remind me to emphasize the importance of units in the reporting of your results and conclusions. Otherwise the answer would be ambiguous.

Thank you for your question and I hope this helps!

-MJM