+816 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.
+128460 Number of band members in first fromation is
(n -2)(n + 8)
Number of band members in second formation = n (2n - 3)
But...the difference in the numbers will be ≥ 4...since at least 4 drummers are excluded in the second grouping....so we have this inequality
(n - 2) ( n + 8) - n (2n - 3) ≥ 4 simplify
n^2 + 6n - 16 - 2n^2 + 3n - 4 ≥ 0
-n^2 + 9n - 20 ≥ 0 multiply through by -1....change the inequality sign direction
n^2 - 9n + 20 ≤ 0 (1)
The easiest way to solve this is to change it to an equality
n^2 - 9n + 20 = 0 factor
(n - 5) ( n - 4) = 0
Setting each factor to 0 and solving for n produces n = 5 or n = 4
We have 3 possible intervals that will solve (1)
(-inf, 4) U [4 ,5 ] U (5 ,inf)
If we let n = 4.5...then note that
(4.5)^2 - 9(4.5) + 20 = -.25 which makes (1) true
So since n can only assume integer values...... n = 4 or n = 5 will make (1) true
And the sum of these values = 9
Here's a graph of the solution : https://www.wolframalpha.com/input/?i=n%5E2+-+9n+%2B+20%C2%A0%E2%89%A4+0
