Question

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

Answer 1

Your answer is going to be 13

Answer 2

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=(1)/(2)`

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

`y = ((1)/(2)^2) + (1)/(2) + 20`

Take common denominator

`y= (81)/(4)`

So our maximum vertical distance is
`(81)/(4)` at
`x= (1)/(2)`