Answer:
k=30
Step-by-step explanation:
The number of mice vary inversely with cats. This is written as:
[tex]Mice \propto \dfrac{1}{Cat} \\Mice = \dfrac{k}{Cat}[/tex]
When there are 3r-19 cats, there are 2r+1 mice
[tex]2r+1 = \dfrac{k}{3r-19}\\$Cross multiply$\\k=(3r-19)(2r+1)[/tex]
When there are 6r-27 mice, there are r-5 cats.
[tex]6r-27 = \dfrac{k}{r-5}\\$Cross multiply$\\k=(6r-27)(r-5)[/tex]
Taking the values of k above, we have:
[tex]k=(3r-19)(2r+1) =(6r-27)(r-5)\\(3r-19)(2r+1) =(6r-27)(r-5)\\6r^2+3r-38r-19=6r^2-30r-27r+135\\$Collect like terms\\6r^2-6r^2+3r+30r-38r+27r-19-135=0\\22r-154=0\\22r=154\\$Divide both sides by 22\\r=7[/tex]
Therefore:
[tex]k=(3r-19)(2r+1) \\=(3*7-19)(2*7+1)\\=(21-19)(14+1)\\=2*15\\k=30[/tex]
