First, let's find how many cups of yogurt the machine fills per hour.
[tex] \dfrac{\frac{3}{4}\, cup}{1 \,tub} \cdot 1600 \,tubs = 1200\, cups [/tex]
Now, we can use our cups-to-gallons conversion to find how many gallons this would be:
[tex] 1200 \,cups \, \cdot \dfrac{1 \,gallon}{16\, cups} = 75 \,gallons [/tex]
Our answer is 75 gallons.