What is the maximum vertical distance between the line y = x + 30 and the parabola y = x2 for −5 ≤ x ≤ 6?

Question

What is the maximum vertical distance between the line  y = x + 30  and the parabola  y = x2 for −5 ≤ x ≤ 6?

anonymous · Answer

vertical distance implies maximize $|y_1-y_2|$ where $y_1=x+30\text{ and }y_2=x^2$

anonymous · Answer

I plug in y1 and y2 to |y1-y2|?

anonymous · Answer

what class are you in?  are you taking calculus?

anonymous · Answer

Yes I'm taking calculus

anonymous · Answer

so $x+30 - x^2$ is the function you want to maximize in the interval $5 \le x \le 6$, right?

anonymous · Answer

Yes

anonymous · Answer

and $x+30 \ge x^2$ in the interval $-5\le x\le 6$ right?

so take the derivative set it equal to 0 and see if that x is in your interval.  if it is, then that's the max (you should check to make sure it is a max but it will be).

anonymous · Answer

once you know the value of x, plug it into the distance function $\left(-x^2+x+30\right)$ and then you'll have the max distance.

anonymous · Answer

I found my inverse which is: -2x+1 right? And I equal it to zero and I got x=1/2
Then I plugged it in the formula and I got 30.75?

anonymous · Answer

there you go! except it should be 30.5 - .25 = 30.25