C ("ate") and D ("eat") are both correct answers which have different meanings. Although they appear to be past and present tense, they are something quite different, "Past Subjunctive Mood" (C) and "Present Subjunctive Mood" (D). The difference between the two has nothing to do with time. Both sentences are timeless. The Present Subjunctive Mood is often simply called "Subjunctive Mood". Present Subjunctive is used to express wishes and conditions contrary to fact. Thus "I'd rather (that) you eat the sandwich in the kitchen" expresses a wish. This sentence implies neither that the person eats in the kitchen nor that he does not eat in the kitchen. It simply expresses a preference for eating in the kitchen. Past Subjunctive is used to express hypothetical or imagined conditions. "I'd rather you ate the sandwich in the kitchen" expresses that eating in the kitchen is an imagined situation. This sentence is especially useful if the person does not eat in the kitchen but you would rather that he did. Since he doesn't eat there now, the imagined situation where he would eat there is the one you prefer. "Ate" is also useful if you are helping the person choose where to eat. Imagine this conversation: X: I want to eat in the living room! Y: I don't want you eating in the living room. I'd rather that you ate in the kitchen. That's a tricky one to understand because Present Subjunctive would also work. It sounds different though. "Ate" is more polite. If Y says "I'd rather that you eat in the kitchen." then Y sounds bossy. It would be more courteous to say "ate" because instead of telling X where Y wants him to eat, Y expresses an imagined place where he would rather that X eat.

ate

C - the subjunctive mood!