As cost of producing #n# items is #950+63n#
As price of one item (if #n# are sold) is #280-5n# and
selling price of #n# items is #280n-n^2#
Hence for profit, we should have
#280n-n^2 >= 950+63n#
or #n^2+63n-280n+950 <= 0#
or #n^2-217n+950 <= 0#
i.e. if #alpha# and #beta# are roots of #n^2-217n+950=0# where #alpha < beta#.
and then #(n-alpha)(n-beta)<=0#
Roots of #n^2-217n+950=0# using quadratic formula are
#n=(217+-sqrt(217^2-4xx950))/2#
or #(217+-208.06)/2# i.e. #212.53# or #4.47#
and so #beta=212.53# or #alpha=4.47#
As #(n-alpha)(n-beta)<=0#
we have either #n-alpha>=0# and #n-beta<=0# i.e. #n>=alpha# and #n<=beta# i.e. #alpha <= n <= beta#
or #n-alpha<=0# and #n-beta>=0# i.e. #n<=alpha# and #n>=beta#, which is not possible as #alpha < beta#.
Hence answer is #4.47<= n <= 212.53# but as we should have only integers answer is #5 <= n <= 212#
