Question

What is the maximum vertical distance between the line y = x + 20 and the parabola y = x2 for −4 ≤ x ≤ 5?

Answer 1

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 = $x+20 -x^2$

y = $-x^2 + x + 20$

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

$x=\frac{1}{2}$

Now we plug in x values in $y= -x^2 + x + 20$

$y = (\frac{1}{2}^2) + \frac{1}{2} + 20$

Take common denominator

$y= \frac{81}{4}$

So our maximum vertical distance is $\frac{81}{4}$ at $x= \frac{1}{2}$

Answer 2

Your answer is going to be 13