The correct option which resembles with the given graph is \(\fbox{\begin\\\ optionA and optionC\\\end{minispace}}\).
Further explanation:
From the given graph the it is observed that the interval is \([8,11)\) which is a half open interval or it can be said that it is a half closed interval.
A half closed or a half open interval is an interval in which one end point of the interval is included but the other point is not included.
For example:
In an interval of the form \([a,b)\) the point \(a\) is included in the set but the point \(b\) is not included in the set.
Similarly, in the interval \([8,11)\) the point \(8\) is included in the set but the point \(11\) is not included in the set.
In order to determine the correct option which resembles with given graph form interval for each option.
Option A:
As per the statement in option A the age of the student must be at least \(8\) years and not more than \(11\) years that is the age of the student is either \(8\) years or greater than \(8\) years and less than \(11\) years.
As per the above statement the interval formed is \(\fbox{\begin\\\ [8,11)\\\end{minispace}}\) in which \(8\) is included in the set and \(11\) is not included which means it is a half closed or half opoen interval.
Therefore, \(\fbox{\begin\\\ optionA\\\end{minispace}}\) is a correct option.
Option B:
As per the statement in option B the amount of money the babysitter can earn is between \(\$8\) and \(\$11\) per hour.
As per the above statement the interval formed is \(\fbox{\begin\\\ (8,11)\\\end{minispace}}\) in which both of the end points are not included in the set which means that it is an open interval.
Therefore, \(\fbox{\begin\\\ optionB\\\end{minispace}}\) is an incorrect option.
Option C:
As per the statement in option C the time required to complete a program in college cannot be less than \(8\) hours and the time must be less than \(11\) hours.
From the above statement it is concluded that the time required can be \(8\) hours or more than \(8\) hours but it has to be less than \(11\) hours.
So, as per the above the statement the interval formed is \(\fbox{\begin\\\ [8,11)\\\end{minispace}}\).
Therefore, \(\fbox{\begin\\\ optionC\\\end{minispace}}\) is a correct option.
Option D:
As per the statement in option D the age of the dogs is either \(8\) years old or below \(8\) years and age of the dog must be greater than \(11\) tears.
As per the above statement the interval formed is \(\fbox{\begin\\\ (-\infty,8) \bigcup(11,\infty)\\\end{minispace}}\).
Therefore, the \(\fbox{\begin\\\ optionD\\\end{minispace}}\) is an incorrect option.
So, \(\fbox{\begin\\\ optionA and optionC\\\end{minispace}}\) are the correct options which resembles with the given graph.
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Answer details:
Grade: High school
Subject: Mathematics
Chapter: Sets
Keywords: Interval, set, open interval, closed interval, half open interval, half closed interval, set theory, program, community college, babysitters, veterinary clinic, ages, students.
