The minimum number of bicycles that must be produced and sold to break even = 20
Explanation:The profit generated by producing and selling n bicycles per week is given by the formula:
[tex]P(n)=-5n^2+400n-6000[/tex]To get the minimum number of bicycles that must be produced and sold to break even, let P(n) = 0 (Since break even means no profit is made)
[tex]\begin{gathered} 0=-5n^2+400n-6000 \\ 5n^2-400n+6000=0 \end{gathered}[/tex]Solve the resulting quadratic equation above
[tex]\begin{gathered} 5n^2-300n-100n+6000=0 \\ 5n(n-60)-100(n-60)=0 \\ (5n-100)(n-60)=0 \\ 5n-100=0 \\ 5n=100 \\ n=\frac{100}{5} \\ n=20 \\ n-60=0 \\ n=60 \end{gathered}[/tex]This means that, to break even, either 60 or 20 bicycles must be produced and sold
The minimum number of bicycles that must be produced and sold to break even = 20
