1. The range of a list of numbers is the difference between the largest number and the smallest number in the list. For example, the range of the list $1,5,1,6,3$ is $6-1=5$.
(a) Determine the range of the list $7,13,4,9,6$.
(b) The list $11,5, a, 13,10$ has a range of 12 . Determine the two possible values of $a$.
(c) The list $6+2 x^{2}, 6+4 x^{2}, 6,6+5 x^{2}$ has a range of 80 . Determine the two possible values of $x$.
(d) The list $5 x+3 y, 0, x+y, 3 x+y$ has a range of 19 . If $x$ and $y$ are integers with $x>0$ and $y>0$, determine the values of $x$ and $y$.
1. (a) We arrange the list in increasing order and obtain $4,6,7,9,13$.
Therefore, the range of this list is $13-4=9$.
(b) Removing $a$ from the list, the smallest number remaining is 5 and the largest number remaining is 13 .
Since $13-5=8$ which is smaller than 12 (the actual range), then $a$ must be either the smallest number or the largest number in the list.
If $a$ is the smallest number in the list, then 13 is the largest and so for the range to be 12 , we must have $13-a=12$ and so $a=1$. Note that if $a=1$, then the list is $1,5,10,11,13$, which has range 12 .
If $a$ is the largest number in the list, then 5 is the smallest and so for the range to be 12 , we must have $a-5=12$ and so $a=17$. Note that if $a=17$, then the list is $5,10,11,13,17$, which has range 12 .
Therefore, the two possible values of $a$ are 1 and 17 .
(c) Since $x^{2} \geq 0$, then $6 \leq 6+2 x^{2}$ and $6+2 x^{2} \leq 6+4 x^{2}$ and $6+4 x^{2} \leq 6+5 x^{2}$.
In other words, when the given list is arranged so that each term is greater than or equal to the one before it, we obtain
$$
6,6+2 x^{2}, 6+4 x^{2}, 6+5 x^{2}
$$

Since the range of this list is 80 , then $\left(6+5 x^{2}\right)-6=80$ and so $5 x^{2}=80$ or $x^{2}=16$, which means that $x= \pm 4$.
(d) Since $x>0$ and $y>0$, then $00$, then $5 x<19$ which means that $x=1, x=2$ or $x=3$. We can check that when $x=1$ and $x=3$, the value of $y$ is not an integer.)
Thus, $x=2$ and $y=3$.