Answer:
B: 27
Step-by-step explanation:
The total number of animals in a shelter is modeled by the function:
[tex]f(x)=2x^2-45x[/tex]
Where x is the number of cages at the facility.
We want to determine the number of cages if there are a total of 243 animals at the shelter.
So, we can set our function to 243 and solve for x:
[tex]243=2x^2-45x[/tex]
Subtract 243 from both sides:
[tex]2x^2-45x-243=0[/tex]
Solve the quadratic. Factoring and completing the square seems too difficult, so we can use the quadratic formula:
[tex]\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
In this case, a = 2, b = -45, and c = -243. Substitute:
[tex]\displaystyle x=\frac{-(-45)\pm\sqrt{(-45)^2-4(2)(-243)}}{2(2)}[/tex]
Evaluae:
[tex]\displaystyle x=\frac{45\pm\sqrt{3969}}{4}[/tex]
Evaluate:
[tex]\displaystyle x=\frac{45\pm63}{4}[/tex]
Therefore, our two solutions are:
[tex]\displaystyle x=\frac{45+63}{4}=27\text{ or } x=\frac{45-63}{4}=-\frac{9}{2}[/tex]
Since we cannot have negative or half a cage, we can ignore the second solution.
Therefore, there are a total of 27 cages at the shelter.
Our answer is B.
