Question

# Perform the inverse transformation to express light intensity as an exponential function of depth in the lake. Display a scatterplot of the original data with exponential model superimposed. Is your exponential function a satisfactory model for the data?

Exponential growth and decay
Perform the inverse transformation to express light intensity as an exponential function of depth in the lake.
Display a scatterplot of the original data with exponential model superimposed. Is your exponential function a satisfactory model for the data?

2020-11-17

Step 1
Equation found in exercise 5d (where $$\displaystyle\hat{{{y}}}$$ is the natural logarithm of the light intensity):
$$\displaystyle{\left[\hat{{{y}}}={6.7891}-{0.330}{x}\right]}$$
Take the exponential of both sides:
$$\displaystyle{\left[\hat{{{y}}}={e}^{{{6.7891}-{0.330}{x}}}={e}^{{{6.7891}}}{e}^{{-{0.330}{x}}}\right]}$$
Step 2
The Depth is on the vertical axis and the light intensity is on the horizontal axis.

Step 3
The model is an satisfactory model, because it seems to pass through every single data point.
Result:
Yes.