Related to triangle bounded by lines

Question

triangle bounded by lines y=0, y=2x and y=(-x/2)+kwherek is positive. Find the value of k knowing that the area of the triangle is 80

Answer 1

Two of the lines forming the triangle are fixed (y=0, y=2x), while one is variable.  One of the vertices is thus fixed at the intersection of y=0 and y=2x, which is the origin.  The remaining vertices occur where y=2x intersects y= -(x/2) + k, and also where y = -(x/2) + k intersects y=0.  Let's call these A and B, respectively.

 

The base length of the triangle is the x-coordinate of B, and the height is given by the y-coordinate of A.  Make a sketch, including y=0, y=2x, and the remaining line for a few positive values of k.  Everything stated above will become crystal clear if you do. 

 

To find the x-coordinate of A, set 2x = -(x/2) + k and solve for x: 

(2+1/2)x = k

(5/2)x = k

x= (2/5)k

 

Since we need the y-coordinate of A as the height of the triangle, substitute this into either equation.  y=2x provides the least difficulty:  y = 2*(2/5)k = (4/5)k.

 

To find the x-coordinate of B, set y = -(x/2) + k to 0 and solve for x:

-(x/2) + k = 0

-(x/2) = -k

x = 2k

 

So we have as the height of the triangle (4/5)k, and as the base length 2k.  Substitute into the equation for the area of a triangle, and set this area to 80:

 

(1/2)b*h = A

(1/2)(2k)(4/5)k = 80

(4/5) k^2 = 80

k^2 = 400/4 = 100

k = ± 10.  Since the problem states that k > 0, retain k = 10.