William, a logistician, needs to route a freight train that is 20 feet at its tallest point and 10 feet at its widest point within 3 days. The most direct path includes a single-track tunnel that needs 24 hour notice prior to use. If the tunnel is roughly modeled by f(x)=-0.1x^2+3.2x- 3.5, should william make arrangements for the train to use the tunnel?

Question

anonymous · Answer

Are you familiar with how to find the vertex of a parabola?

anonymous · Answer

No

anonymous · Answer

Are you familiar with what a vertex is?

anonymous · Answer

lowest or highest point on a parabola

anonymous · Answer

Righty-o.  So in this case, it's the highest point because it looks like this:
|dw:1375467265549:dw|
We want to know if the highest point on the parabola is at least 20.
The x-coordinate of the vertex is located at
$$x = \frac{-b}{2a}$$
Let me know if/when you find the x-coordinate of the vertex or if you need help doing that.

anonymous · Answer

i just wanna know how to solve the equation f(x)=-0.1x^2+3.2x-3.5

anonymous · Answer

Okidoki, we can switch over to the width part.  You want to know how wide this thing is:
|dw:1375467595119:dw|
So basically you need to find the location between those two points and find the distance between them.  Use the quadratic formula on the equation
$$0=-0.1x^2+3.2x-3.5$$
Quadratic formula:
$$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$