The vertical distance can be found by subtracting parabola from the given line.
Given line equation is y= x+20
Equation of parabola is y= x^2
Subtract parabola from line equation
so y = [tex] x+20 -x^2 [/tex]
y = [tex] -x^2 + x + 20 [/tex]
Now we take derivative to find out the maximum that is the vertex
y' = -2x + 1
Now we set derivative =0 and solve for x
0 = -2x+1
2x = 1
[tex] x=\frac{1}{2} [/tex]
Now we plug in x values in [tex] y= -x^2 + x + 20 [/tex]
[tex] y = (\frac{1}{2}^2) + \frac{1}{2} + 20 [/tex]
Take common denominator
[tex] y= \frac{81}{4} [/tex]
So our maximum vertical distance is [tex] \frac{81}{4} [/tex] at [tex] x= \frac{1}{2} [/tex]
