# A band is marching in a rectangular formation with dimensions n-2 and n + 8. In the second stage of their performance, they re-arrange to form a different rectangle with dimensions n and 2n - 3, excluding all the drummers. If there are at least 4 drummers, then find the sum of all possible values of n?

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Answer:The sum of all possible values of n is 9.Step-by-step explanation:We are going to solve this problem by subtracting areas.For the first stage, the rectangular area of the formation is :[tex](n-2).(n+8)[/tex]In the second stage, the rectangular area of the formation is :[tex]n(2n-3)[/tex]We know that in this second formation they excluded all the drummers and also we know that there are at least 4 drummers.Therefore, the difference between the areas of the first and the second formation is :[tex](n-2).(n+8)-n.(2n-3)[/tex] and this area must be at least 4 (because of the drummers excluded)[tex](n-2).(n+8)-n.(2n-3)\geq 4[/tex][tex]n^{2}+8n-2n-16-2n^{2}+3n\geq 4[/tex][tex]-n^{2}+9n-16\geq 4[/tex][tex]-n^{2}+9n-20\geq 0[/tex] (I)We need to solve this and find the possibles ''n'' that satisfy the inequality.First we look for the values that satisfy[tex]-n^{2}+9n-20=0[/tex] (II)Using the quadratic equation :[tex]n_{1}=4\\n_{2}=5[/tex]For this values of ''n'' the inequality (I) is satisfied.Now we study the vertex.Given a quadratic function [tex]f(x)=ax^{2}+bx+c[/tex]The coordinate ''x'' of the vertex is [tex]\frac{-b}{2a}[/tex]For (II) [tex]a=-1\\b=9\\c=-20[/tex][tex]\frac{-b}{2a}=\frac{-9}{2(-1)}=\frac{9}{2}=4.5[/tex]This is the coordinate ''x'' of the vertex.For the coordinate ''y'' we calculate [tex]f(xVertex)[/tex][tex]f(4.5)=-(4.5)^{2}+9(4.5)-20=0.25[/tex]That is positive. The coordinates of the vertex are [tex](4.5,0.25)[/tex]In the quadratic function [tex]a=-1\\a<0[/tex]So it is a negative quadratic function.We conclude that for the interval [4,5] the quadratic function is positive, therefore between [4,5] the inequality (I) is satisfied.The two possible values for n are 4 and 5.Finally, [tex]4+5=9[/tex] is the sum of all possible values of n(Notice that n must be an integer number)
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