There are 0 intersections between the railroad and the highway and there are 2 intersections between the railroad and the turnpike
The number of intersections there will be between the railroad and the highway
From the graph, we have the following points on the railroad
(x, y) = (3, 3) and (0, 5)
The slope is calculated as:
m = (y2 - y1)/(x2 - x1)
This gives
m = (5 -3)/(0 - 3)
Evaluate
m = -2/3
The equation is then calculated as:
y = mx + b
This gives
y = -2/3x + b
Substitute (0, 5)
5 = -2/3 * 0 + b
This gives
b = 5
Substitute b = 5 in y = -2/3x + b
y = -2/3x + 5
Multiply through by 3
3y = -2x + 15
Rewrite as:
2x + 3y = 15
The equation of the highway's path is
2x+3y=21
So, we have:
2x+3y=21
2x + 3y = 15
Subtract the equations
2x - 2x + 3y - 3y = 21 - 15
Evaluate
0 = 6
The above equation is false because 0 and 6 are not equal.
This means that the system of equations have no solution
Hence, there are 0 intersections between the railroad and the highway.
How many intersections there will be between the railroad and the turnpike?
Here, we have
y = x^2 and 2x + 3y = 15
Substitute y = x^2 in 2x + 3y = 15
2x + 3x^2 = 15
Rewrite as:
3x^2 + 2x - 15 = 0
Calculate the discriminant using
d =b^2 - 4ac
This gives
d = 2^2 - 4 * 3 * -15
Evaluate
d = 184
The discriminant d is greater than 0.
This means that the equation has 2 real solutions
Hence, there are 2 intersections between the railroad and the turnpike
Read more about linear and quadratic functions at:
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