Fish scaling
You have caught eight individuals of a particular fish species and measured the length, depth (vertical height), and thickness (side-to-side distance) of each. For each fish, approximate total surface area and volume of each fish were calculated, as shown in the table below.
|
LENGTH (cm)
|
THICKNESS (cm)
|
DEPTH (cm)
|
VOLUME (cm3)
|
AREA (cm2)
|
|
2
|
0.5
|
1
|
1
|
6
|
|
4
|
1
|
2
|
8
|
24
|
|
6
|
1.5
|
3
|
27
|
54
|
|
10
|
2.5
|
5
|
125
|
150
|
|
16
|
4
|
8
|
512
|
384
|
|
24
|
6
|
12
|
1728
|
864
|
|
48
|
12
|
24
|
13824
|
3456
|
|
60
|
15
|
30
|
27000
|
5400
|
Use Excel (or a similar program) to fit power functions (scaling equations, Y=axb) to these data (a power function, which has the form y = axb, has a variable base raised to a fixed exponent). Exactly how to do this will depend on your version of Excel. But the procedure will be something like this:
1. Select the two columns of numbers you wish to plot (e.g., depth and volume).
2. Click on the Charts tab & select Scatter plot
3. Select the resulting plot and click on the Chart Layout tab & Trendline subtab.
4. Under Trendline Options, select Power & under Options (just to the left) click on "Display equation on chart" and click OK.
The equation of the power function fitted to your data should be displayed on your graph.
1a) What is the value of b for depth vs. volume (note that "A versus B" tells you that A is on the Y axis and B is on the X axis)? Paste your graph below.
1b) What is the value of b for area vs. volume? Paste your graph below.
1c) What is the value of b for volume vs. area? Paste your graph below.
1d) What is the volume of b for thickness vs. area? Paste your graph below.
1e) Do your fish appear to be growing isometrically or allometrically? Explain.