Which of the following would linearize the data for volume and time?

Question

Water is poured into a large, cone-shaped cistern. The volume of water, measured in cm³, is reported at different time intervals, measured in seconds. The scatterplot of volume versus time showed a curved pattern.

Which of the following would linearize the data for volume and time?

A.Seconds, cm³

B.ln(Seconds), cm³

C.Seconds, ln(cm³)

D.ln(Seconds), ln(cm³)

Answer 1

To linearize the data for volume and time, we need to find a transformation that will make the relationship between volume and time linear. A common transformation for nonlinear relationships is to take the logarithm of one or both variables.

To determine which transformation will linearize the data, we can look at the form of the relationship between volume and time in the scatterplot. Since the pattern is curved, it suggests that the relationship may be exponential. Therefore, we can try taking the logarithm of the volume or the time or both and see which transformation produces a linear relationship.

A) Seconds, cm3: This transformation does not involve taking the logarithm of either variable, so it is unlikely to linearize the relationship.

B) ln(Seconds), cm3: This transformation takes the natural logarithm of the time variable. It may help to linearize the relationship if the relationship is exponential with respect to time.

C) Seconds, ln(cm3): This transformation takes the natural logarithm of the volume variable. It is unlikely to linearize the relationship because it does not address the potential exponential relationship with respect to time.

D) ln(Seconds), ln(cm3): This transformation takes the natural logarithm of both variables. It is a good choice because it can linearize an exponential relationship between the two variables.

Therefore, the transformation that would linearize the data for volume and time is D) ln(Seconds), ln(cm3).