We know that six members of the math club are forming two teams for a contest and the teams will be determined by having each student draw a number from a box. We are given a table that shows the result of the draw.
Student
Number Drawn
Lydia
sqrt(38)
Marcy
6.3 4
Caleb
sqrt(36)
Ryan
6.343443444...
Anya
6.34
Chan
sqrt(34)
We are told that the students who drew rational numbers will form the team called the Tigers. The students who drew irrational numbers will form the team called the Lions. We are asked to list the members of each team. To do so, we will begin by recalling the definition a ration number.
Rational Number
Any number that can be written as the quotient of two integers is called a rational number. A rational number can be written in the form ab or - ab, where a and b are integers and b ≠ 0. A rational number can be a whole number, a fraction, or a decimal.
Now we will recall the definition of an irrational number.
Irrational Number
An irrational number is a number that cannot be written in the form ab, where a and b are integers and b ≠ 0.
To list the members of each team, we will determine whether each student drew a rational or irrational number.
Lydia
The number Lydia drew is sqrt(38). We can see that it is a square root of a whole number. Recall that the square root of a perfect square is a rational number. However, for any whole number b that is not a perfect square, sqrt(b) is irrational. Let's list the first few perfect squares.
4,9,16,25,36,49
As we can see, 38 is not a perfect square. Therefore, sqrt(38) is an irrational number, so Lydia is on the Lions team.
Marcy
The number that Marcy drew is 6.34. We already know that repeating decimals are rational numbers, so Marcy drew a rational number. She is on the Tigers team.
Caleb
Caleb drew the number sqrt(36). Notice that 36 is a perfect square, and we know that the square root of a perfect square is a rational number. Therefore, Caleb drew a rational number and is on the Tigers team.
Ryan
The number Ryan drew is 6.343443444 ..., which we can see is a number with decimal expansion that does not repeat or terminate. Therefore, it is an irrational number and Ryan is a Lion.
Anya
Anya drew the number 6.34. It is a repeating decimal, so it is a rational number. Anya will join the Tigers team.
Chan
Finally, Chan drew sqrt(34). Because the number 34 is not a perfect square, sqrt(34) is an irrational number, and Chan is the final member on the Lions team.
Conclusion
Now we can list the members of each team. The students who drew rational numbers form the team called the Tigers and the students who drew irrational number for the team called the Lions.
Tigers
Lions
Marcy
Lydia
Caleb
Ryan
Anya
Chan
We are told that the student on each team who drew the greatest number will be the captain of that team. We want to determine who the captain of the Tigers will be. Let's take a look at the numbers drawn by the members of this team.
Tigers
Number Drawn
Marcy
6.3 4
Caleb
sqrt(36)
Anya
6.34
Let's start by evaluating sqrt(36). We know that 36 is a perfect square.
sqrt(36) = 6
We can tell that 6 is less than the numbers Marcy and Anya drew. Now let's approximate 6.3 4 and 6. 34.
&6.3 4 = 6.34444444 ... ≈ 6.344
&6. 34 = 6.34343434 ... ≈ 6.343
We can see that 6.34 is less than 6.3 4. This means that 6.3 4 is the greatest number, which makes Marcy the captain of the Tigers.
Now we want to find out who will be the captain of the Lions. We again need to find the member of this team who drew the greatest number.
Lions
Number Drawn
Lydia
sqrt(38)
Ryan
6.343443444...
Chan
sqrt(34)
The number 34 is less than 38, so we can conclude that Chan will not be the captain of the Lions. This is because Chan's number sqrt(34) is less than sqrt(38). Let's compare 6.343443444... and sqrt(38). We will approximate sqrt(38) using perfect squares. The two closest perfect squares to the number 38 are 36 and 49.
36 < 38 < 49
Now we can rewrite the above inequality with square roots. We will also calculate the square roots of the perfect squares.
sqrt(36) < sqrt(38) < sqrt(49)
⇕
6 < sqrt(38) < 7
We found that sqrt(38) is somewhere between 6 and 7. However, 6.343443444... is also between 6 and 7, which means that we need a better approximation. To do it, we will use decimals. Let's see if sqrt(38) is greater than 6.3.
6.3 * 6.3 = 39.69
6 < sqrt(38) < 6.3
Now we know that sqrt(38) is greater than 6 but less than 6.3. Because 6.343443444... is greater than 6.3, we get that 6.343443444... is the greatest number. This means that Ryan is the captain of the Lions.
