Final answer:
The constraints for a thriving population of gazelles at the zoo, with a limit of 12 female gazelles and the need for thrice as many males, are 0 < x ≤ 12 and 0 < y ≤ 36, which matches answer option B.
Step-by-step explanation:
To represent a thriving population of gazelles in the zoo, we must write constraints that capture the relationship between the number of female gazelles (x) and the number of male gazelles (y). Since the zoo can only accommodate 12 female gazelles, and the herd requires three times more males than females, we can express the constraints as inequalities.
The first constraint pertains to the maximum number of female gazelles, which is 12. Therefore, we have 0 < x ≤ 12. The second constraint relates to the male gazelles, as we need three times as many males as there are females, so y = 3x. Since x has an upper limit of 12, the maximum value for y would be 36 (3 times 12).
Taking into account that we cannot have a negative number of gazelles, the constraints for a population to thrive are:
- 0 < x ≤ 12: the number of female gazelles must be positive and no more than 12.
- 0 < y ≤ 36: the number of male gazelles must be positive and no more than three times the maximum number of female gazelles.
Given these constraints, the correct answer is B. 0 < x ≤ 12 and 0 < y ≤ 36.