To solve this problem, we need to establish the constraints for the number of male and female gazelles that can be accommodated at the zoo, given the specific conditions.
1. Condition on the Number of Female Gazelles:
The zoo can house a maximum of 12 female gazelles. This can be translated mathematically to:
[tex]\[
0 < x \leq 12
\][/tex]
Here, [tex]\(x\)[/tex] must be greater than 0 because we need a positive number of female gazelles and can be at most 12.
2. Condition on the Number of Male Gazelles:
The herd needs to have exactly three times more males than females. Therefore, if [tex]\(x\)[/tex] represents the number of female gazelles, then the number of male gazelles, [tex]\(y\)[/tex], is given by:
[tex]\[
y = 3x
\][/tex]
This ensures that the number of males is always three times the number of females.
3. Implication on the Maximum Number of Male Gazelles:
Since [tex]\(x \leq 12\)[/tex], substituting into [tex]\(y = 3x\)[/tex] gives the maximum number of male gazelles as:
[tex]\[
y = 3 \cdot 12 = 36
\][/tex]
Hence, the constraint on [tex]\(y\)[/tex] needs to ensure that [tex]\(y\)[/tex] is positive and does not exceed 36:
[tex]\[
0 < y \leq 36
\][/tex]
4. Combining Both Constraints:
We need to combine the constraints on both [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. Therefore, we have:
[tex]\[
0 < x \leq 12 \quad \text{and} \quad 0 < y \leq 36
\][/tex]
The correct options provide constraints on the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. Among the given choices:
a. [tex]\(x > 0\)[/tex] and [tex]\(y > 0\)[/tex]
This choice does not specify the upper bound for [tex]\(x\)[/tex] and [tex]\(y\)[/tex], so it isn’t correct.
b. [tex]\(0 < x \leq 12\)[/tex] and [tex]\(0 < y \leq 36\)[/tex]
This correctly sets the bounds for both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] according to our derived values.
c. [tex]\(0 < x \leq 12\)[/tex] and [tex]\(y > 36\)[/tex]
This does not satisfy the condition that [tex]\(y \leq 36\)[/tex], so it isn’t correct.
d. [tex]\(x > 0\)[/tex] and [tex]\(y < 23\)[/tex]
This sets an inappropriate upper bound on [tex]\(y\)[/tex], so it isn’t correct.
The correct answer is:
b. [tex]\(0 < x \leq 12\)[/tex] and [tex]\(0 < y \leq 36\)[/tex]
