To this point in the lesson, you have learned how to find an exponential regression function for a set of data and determine if the exponential model is appropriate for the data. The remainder of this lesson focuses on using the exponential regression function to solve problems.

Recall from Lesson 6B that real world phenomena can be modelled by functions that describe how things grow or decay as time passes. Examples of such phenomena include the studies of populations, bacteria, the AIDS virus, radioactive substances, electricity, temperatures, and credit payments.

Example 4 demonstrates how a real world problem can be solved using the exponential regression function as a model.

The intensity of light below the surface of a particular lake is reduced by 4% for every metre below the surface. The table below gives the intensity of light at various depths. Model this data with an exponential regression function and use your model to answer the following questions.

  1. What percent of the original intensity of light remains at 10 metres below the surface?
  2. How far below the surface must the light travel before its intensity is 30% of the surface intensity?

Depth (m)
0
2
4
8
12
16
20
24
Percent of Original Light Intensity
100
92.16
84.93
72.14
61.27
52.04
44.20
37.54

Model the data on the graphing calculator using Depth as the x-value and Percent of Original Light Intensity as the y-value.

The exponential regression model is y = 100(0.96) x.

  1. To find the percent of original light intensity when the depth is 10 metres means to find y when x = 10.

Step 1: Go to the CALCULATE menu; press 2nd, TRACE.

Step 2: Type in the required x-value; press 1.

Step 3: Type in the required x-value; press 10.

Step 4: Get the corresponding y-value; press ENTER.

At a depth of 10 m, about 66% of the original light intensity remains.

  1. To find the distance below the surface when the light intensity is 30% of the original means to find x when y = 30.

Step 1: Input the given y-value; press Y=, down arrow to scroll to Y2, type in 30.

Step 2: Graph the functions; press GRAPH.

Step 3: Go to the CALCULATE menu; press 2nd, TRACE.

Step 4: Find the intersection of Y1 and Y2; press 5, ENTER, ENTER, ENTER.

The light is reduced to 30% of the surface intensity at a depth of about 29.5 m.