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Choice C is correct. It’s given that the equation $3x+5y=32$ represents the situation where Keenan filled $x$ small jars and $y$ large jars with all the vegetable broth he made, which was $32$ cups. Therefore, $3x$ represents the total number of cups of vegetable broth in the small jars and $5y$ represents the total number of cups of vegetable broth in the large jars. 

Choice A is incorrect. The number of large jars Keenan filled is represented by $y$, not $5y$.

Choice B is incorrect. The number of small jars Keenan filled is represented by $x$, not $5y$.

Choice D is incorrect. The total number of cups of vegetable broth in the small jars is represented by $3x$, not $5y$.