Final answer:
The secret word is CAT, as it is the only word among the options given where the total number of vowels can be determined by revealing a single letter.
Step-by-step explanation:
To solve the puzzle of determining the secret word, you need to identify which of the given options—CAT, DOG, FOX, RAT—would allow you to know the total number of vowels if just one letter was revealed. Let's analyze the options: CAT contains one vowel. Revealing any letter may indicate the presence or absence of a vowel. DOG contains one vowel as well. However, revealing the 'D' or 'G' would not indicate the number of vowels.
FOX has one vowel, which again does not help ascertain the number of vowels when revealing either 'F' or 'X'. RAT contains one vowel. The same logic as CAT applies here. Considering this, the only word that allows you to know the total number of vowels by revealing just a single letter is CAT.
If the letter 'A' is revealed, you know there is one vowel. However, even if 'C' or 'T' is revealed, because of the position of the known vowel and the structure of the word, you can deduce that there is only one vowel in the word. Hence, the secret word is CAT, represented by option (a).