As cost of producing nn items is 950+63n950+63n
As price of one item (if nn are sold) is 280-5n280−5n and
selling price of nn items is 280n-n^2280n−n2
Hence for profit, we should have
280n-n^2 >= 950+63n280n−n2≥950+63n
or n^2+63n-280n+950 <= 0n2+63n−280n+950≤0
or n^2-217n+950 <= 0n2−217n+950≤0
i.e. if alphaα and betaβ are roots of n^2-217n+950=0n2−217n+950=0 where alpha < betaα<β.
and then (n-alpha)(n-beta)<=0(n−α)(n−β)≤0
Roots of n^2-217n+950=0n2−217n+950=0 using quadratic formula are
n=(217+-sqrt(217^2-4xx950))/2n=217±√2172−4×9502
or (217+-208.06)/2217±208.062 i.e. 212.53212.53 or 4.474.47
and so beta=212.53β=212.53 or alpha=4.47α=4.47
As (n-alpha)(n-beta)<=0(n−α)(n−β)≤0
we have either n-alpha>=0n−α≥0 and n-beta<=0n−β≤0 i.e. n>=alphan≥α and n<=betan≤β i.e. alpha <= n <= betaα≤n≤β
or n-alpha<=0n−α≤0 and n-beta>=0n−β≥0 i.e. n<=alphan≤α and n>=betan≥β, which is not possible as alpha < betaα<β.
Hence answer is 4.47<= n <= 212.534.47≤n≤212.53 but as we should have only integers answer is 5 <= n <= 2125≤n≤212
