Answer :
We start by converting the water tank’s capacity from gallons to ounces. Since one gallon is [tex]$128$[/tex] ounces, a tank of [tex]$5$[/tex] gallons contains
[tex]$$
5 \times 128 = 640 \text{ ounces}.
$$[/tex]
Next, Tom fills [tex]$6$[/tex] bottles, each holding [tex]$16$[/tex] ounces. Thus, the total amount used for the bottles is
[tex]$$
6 \times 16 = 96 \text{ ounces}.
$$[/tex]
Additionally, he fills a pitcher that holds [tex]$\frac{1}{2}$[/tex] gallon. Converting this amount to ounces gives
[tex]$$
\frac{1}{2} \times 128 = 64 \text{ ounces}.
$$[/tex]
Now, the total amount of water used is the sum of the water for the bottles and the pitcher:
[tex]$$
96 + 64 = 160 \text{ ounces}.
$$[/tex]
Finally, to determine the amount of water left in the tank, subtract the total water used from the initial amount:
[tex]$$
640 - 160 = 480 \text{ ounces}.
$$[/tex]
Thus, there are [tex]$\boxed{480}$[/tex] ounces of water left in the tank.
[tex]$$
5 \times 128 = 640 \text{ ounces}.
$$[/tex]
Next, Tom fills [tex]$6$[/tex] bottles, each holding [tex]$16$[/tex] ounces. Thus, the total amount used for the bottles is
[tex]$$
6 \times 16 = 96 \text{ ounces}.
$$[/tex]
Additionally, he fills a pitcher that holds [tex]$\frac{1}{2}$[/tex] gallon. Converting this amount to ounces gives
[tex]$$
\frac{1}{2} \times 128 = 64 \text{ ounces}.
$$[/tex]
Now, the total amount of water used is the sum of the water for the bottles and the pitcher:
[tex]$$
96 + 64 = 160 \text{ ounces}.
$$[/tex]
Finally, to determine the amount of water left in the tank, subtract the total water used from the initial amount:
[tex]$$
640 - 160 = 480 \text{ ounces}.
$$[/tex]
Thus, there are [tex]$\boxed{480}$[/tex] ounces of water left in the tank.