Unit 4, Lesson 3
Understanding Rational Inputs
Integrated Math 3
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A chemical is accidentally spilled into a lake and needs to be cleaned up. The cleaning process decreases the amount of the chemical in the lake roughly exponentially. Here is a graph representing , an exponential function that models the amount of chemical left in the lake, hours after the cleaning begins.
Some exponential functions can have inputs that are any numbers on the number line, not just integers.
Suppose the area of a pond covered by algae , in square meters, is modeled by , where is the number of weeks since a treatment was applied to the pond. How could we use this equation to determine the area covered after 1 day?
Well, since is one week and each week has 7 days, is 1 day. So after 1 day, the algae covers square meters, or about 181 square meters. Using a calculator, we know that the expression , which is equivalent to , is about 0.906. This means that after 1 day, only 91% of the algae from the previous day remains.
This information can also be seen on a graph representing the area. The point at marks the area covered by the algae after 1 week. Point marks the covered area after of a week, or one day.
The graph can be used to estimate the vertical coordinate of and shows that it is close to 180.