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A triangle is formed with edges along the line $y=\frac{2}{3}x+5$, the $x$-axis, and the line $x=k$. If the area of the triangle is less than $20$, find the sum of all possible integral values of $k$.

Guest Feb 28, 2018
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Look at the graph, here...two triangles are possible :

https://www.desmos.com/calculator/m6wnjpgldq

The  height of the triangles at any point will be formed by

[ (2/3)x + 5 ]

And the bases will be  [ x - (- 7.5)]   = [ x + 7.5]

So....we want to solve this

(1/2) [ (2/3)x + 5 ] [ x + 7.5 ]  =  20

[ (2/3)x + 5 ] [ x + 7.5]  =  40

(2/3)x^2 + 5x + 5x  + 37.5  = 0

(2/3)x^2 + 10x - 2.5  =  0

Using a little technology.....the max x  value for the triangle formed above the x axis will be ≈ .246

And the min x value for the triangle formed below the x axis will be ≈ -15.246

With the given boundaries, the  integer sums  of all possible x values of k giving  triangles with an area < 20 units^2  =

[ (-15) + (-14 ) + (-13) + ...+ ( -2) + ( - 1 ) + 0 ]  =

-  (15) (16) / 2   =

-120

CPhill  Feb 28, 2018

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