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

#1**+1 **

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