How fast is the tree growing?

Question

This information is given: Mrs. Fitzgerald planted a tree. After 10 days, the tree measured 39 inches tall. After 28 days, the tree measured 51 inches tall.

This is what I did: Since we are given two points, I found the slope. To find slope between two points, you find the change in y over the change in x. Therefore, the two changes are 28-10 and 51-39.

*UPDATE: I have caught my mistake: for some odd reason, I thought that time was the dependent variable, though I know that it really is the independent. I apologize for my silly mistake :)

What I am confused about: I first wrote down that the change in y is 28-10, since they are the days. In this circumstance, days (time) is the dependent variable. Therefore, they are the y-coordinates. Similarly, 51-39 are the inches, and inches is the independent variable, which makes the 51 and 39 the x-coordinates. However, my friend believes that it is the other way around - 28-10 is the change in x and 51-39 is the change in y; however, I don't understand why this would be.

Who is correct? If my friend is correct, why is that?

Answer 1

Your friend is correct, you can see this by looking at the units of the measurements.

The slope is $\frac{\text{Change in y}}{\text{Change in x}}$

With your friend's approach the slope is calculated as $\frac{51-39\text{ inches}}{28-10\text{ days}} = \frac{12\text{ inches}}{18\text{ days}}$

$ \frac{12\text{ inches}}{18\text{ days}} = \left( \frac{12}{18} \right) \frac{\text{inches}}{\text{days}} = 0.67 \frac{\text{inches}}{\text{days}}$

The value of the slope is $0.67$ and the unit is $\frac{\text{inches}}{\text{days}}$. This matches what we intuitively expect for the speed of tree growth, you should expect to measure the speed in "inches per day", that's why the calculation gave a unit of inches divided by days.

Answer 2

The answer is, it ultimately depends on what you want. The dependent variable is what you are measuring is changing depending on your independent variable.

So if you care how the tree size changes as time passes by, then size is the independent variable (y axis) and time is the dependent variable (x axis).

If however you care how much time passes by as as the tree grows, then time is the independent variable and size is the dependent variable.

In this case, the question implies you care how tree size changes as time passes by, so time would be the independent variable (x axis) and size would be the dependent variable.

Here is a further clarification. You want to know how fast the tree is growing in inches per day i.e. $inches \over day$. The units here imply that you are dividing the change in tree size (inches) by the change in time (days) to get the $inches \over day$ that the tree grows,