Given
[tex]y=\frac{1}{3x^2}[/tex]2x+3y=18
Find
Prove algebraically how many intwrsections there will be between the railroad
Explanation
The graph of 2x+3y=18 is as the picture
2x+3y=18
when x=0, 0+3y=18 => y=6 =>(0,6)
when y=0, 2x+0=18 => x=9 => (9,0)
The intersection between the railroad and the highway is 0 because the graph of the railroad and the graph of the highway are parallel, that means they have no intersection
(b)
Assume the railroad can be found using the equation y=3/2x+b
when x=0 => y=8
[tex]\begin{gathered} \frac{1}{3}x^2=\frac{3}{2}x+8 \\ 2x^2-9x-48=0 \\ D=9^2-4(2)(-48)=465 \\ =>D>0 \\ \frac{1}{3}x^2=\frac{3}{2}x+8 \end{gathered}[/tex]has two roots, and there are 2 intersections
Final Answer
(a) No intersection
(b) Two intersections