Question

a railroad tracks can be determined using the following graph. Several different rosdways are in the same region as the railroad. Part A: A highways path can be found using the equation 2x+3y=18. Use the graphs of the functions to determine the number of the intersections there will he between the railroad and the highway, and explain completely. Part B: A turnpikes route is determined by the equation y=1/3x^2. Prove algebraically how many intwrsections there will be between the railroad abd the turnpike,showing all necessary work

Answer 1

Solution:

Given that the railroad's track is determined using the graph below:

and the highway's path can be found using the equation:

$2x+3y=18$

To determine the intersection, we plot the graph of the highway's path.

To plot the graph, we solve for y for various values of x.

Thus, we have

$\begin{gathered} when\text{ x=0,} \\ 2(0)+3y=18 \\ \Rightarrow3y=18 \\ divide\text{ both sides by 3} \\ \frac{3y}{3}=\frac{18}{3} \\ \Rightarrow y=6 \\ \\ when\text{ x =9,} \\ 2(9)+3y=18 \\ \Rightarrow18+3y=18 \\ subtract\text{ 18 from both sides,} \\ 3y=0 \\ divide\text{ both sides by 3} \\ \frac{3y}{3}=\frac{0}{3} \\ \Rightarrow y=0 \end{gathered}$

By plotting the x and y values as points (x, y) on a graph, we have

From the graphs, we can conclude there are no intersections between the railroad and the highway