Some high school physics students dropped a ball and measured the distance fallen (in centimeters) at various times (in seconds) after its release. If you have studied physics, you probably know that the theoretical relationship between the variables is distance$=490{\left(time\right)}^2$. Which of the following scatterplots would not approximately follow a straight line?

a. A plot of distance versus ${\left(time\right)}^2$

b. A plot of $\sqrt{dis\tan ce}$versus time

c. A plot of distance versus $\sqrt{time}$

d. A plot of ln(distance) versus ln(time)

e. A plot of log(distance) versus log(time)

The theoretical relationship between the variables is distance$=490{\left(time\right)}^2$

a. A graph of distance vs time square would roughly follow a straight line; this is because distance$=490{\left(time\right)}^2$, and so $y=490x$would be the straight line.

b. A graph of $\sqrt{dis\tan ce}$vs time would roughly follow a straight line, because taking the square root of either side of distance$=490{\left(time\right)}^2$results in $\sqrt{dis\tan ce}=\sqrt{490}$time, and so $y=\sqrt{490}x$would be the straight line.

c. A graph of distance versus $\sqrt{time}$would not roughly follow a straight line; this is because the distance$=490{\left(time\right)}^2$, and therefore there is a linear connection between distance and ${\left(time\right)}^2$, but not between distance and $\sqrt{time}$.

d. The pattern in a dotted figure of a power model can be straightened by taking the logarithm of each side. This is because the distance$=490{\left(time\right)}^2$is a power model (as taking the power of time) and it can straighten the pattern in a dotted figure of a power model by taking the logarithm of each side.

e. A graph of log distance versus log time would resemble a straight line. This is because the distance$=490{\left(time\right)}^2$is a power model (as we take the power of time) and it can straighten the pattern in a dotted figure of a power model by taking the logarithm of each side.

The correct option is (c).