Find the restricted domain written in interval notation and interpret what it means.

Question

Environmental scientists often use cost-benefit models to estimate the cost of removing a pollutant from the environment as a function of the percentage of pollutant removed. Suppose a cost benefit function for the cost C (in millions of dollars) of removing x percent of the pollutants from Maple Lake is given by:

 

c(x)= 25x/100-x

 

Find the restricted domain written in interval notation and interpret what it means within the context of the problem. Be sure that you restrict the domain based not only on mathematics, but also on the limitations given in the problem.

Answer 1

 

<I'm going to assume (100-x) is in the denominator because otherwise the domain wouldn't be restricted.

 

c(x)= 25x/(100-x)

 

The domain is all values of x for which the function exists. Since this is a fraction/division you cannot divide by zero, so... the function des not exist where (100 - x) = 0.

 

100 - x ≠ 0

x ≠ 100

 

"x" is the percent of the pollutants to be removed from Maple Lake, so mathematically the function doesn't allow 100% pollutant removal. You could also never remove more than 100% because 100% is all of the pollutants in the lake.

 

x < 100

 

You also can't go below 0% pollutant removal. A negative percent of pollutants might indicate the addition of pollutants to the lake which isn't logical considering the problem. If x = 0 then no pollutants are removed costing nothing. So x = 0 works in this problem.

 

c(0) = 25*0/(100-0) = 0

 

so that gives;

x ≥ 0

 

So the solution in interval notation would be: [0,100)

The square bracket, [ represents greater than or equal to

The rounded parenthesis, ) represents less than