Question

The swinging pendulum Mrs. Hanrahan’s precalculus class collected data on the length (in centimeters) of a pendulum and the time (in seconds) the pendulum took to complete one back-and-forth swing (called its period). Here are their data:

(a) Make a reasonably accurate scatterplot of the data by hand, using length as the explanatory variable. Describe what you see. (b) The theoretical relationship between a pendulum’s length and its period is

$\text{period}=\frac{2\pi }{\sqrt{g}}\sqrt{\text{length}}$

where $g$is a constant representing the acceleration due to gravity (in this case, $g=980\mathrm{cm}/{\mathrm{s}}^2$). Use the graph below to identify the transformation that was used to linearize the curved pattern in part (a).

(c) Use the following graph to identify the transformation that was used to linearize the curved pattern in part (a).

Short answer

a). An outlier in the upper right corner.

b). The used transformation is the square root.

c). The used transformation is the square.

Step-by-step solution

Part (a) Step 1: Given Information

Part (a) Step 2: Explanation

The scatterplot of the following data given in the question taking length as the explanatory variable is as follows:

Thus, by looking at the scatterplot we can say that,

Direction: Positive, because the scatterplot slopes upwards.

Form: Curved, because the point in the upper right corner deviates from the general linear pattern.

Strength: Strong, because all points (except for one) lie very close together.

We also note an outlier in the upper right corner.

Part (b) Step 3: Given Information

Part (b) Step 4: Explanation

The plot is given in the question for part (b) and also it is given that:

$\text{Period}=\frac{2x}{\sqrt{g}}\sqrt{\text{length}}$

This means that the explanatory variable on the horizontal axis is the square root of the length and the response variable on the vertical axis is the period. And this implies that the used transformation is the square root.

Part (c)Step 5: Given Information

Part (c) Step 6: Explanation

The plot is given in the question for part (c) and also it is given that:

$\text{Period}=\frac{2x}{\sqrt{g}}\sqrt{\text{length}}$

$\Rightarrow {\text{Period}}^2=\frac{(2x{)}^2}{g}\text{Length}$

Thus, we note that the values on the horizontal axis are the same as the values for the length while the values on the vertical axis are higher than the values for the period given in the table. This then means that the explanatory variable on the horizontal axis is the length and the response variable on the vertical axis is the square of the period. This then implies that the used transformation is the square.