Hello, rtblue!
What is the maximum height that an 8footwide truck can be in order
to pass through a parabolic tunnel that is 18 feet high and 12 feet wide? Code:
(0,18)
*
*  *
*  *
*  *

*  *


*+*
(6,0)  (6,0)

The general equation of a parabola is: .$\displaystyle y \:=\:ax^2 + bx + c$
The parabola passes through three points: .$\displaystyle (6,0),\:(\text{}6,0),\:(0,18)$
Substitute these coordinates into the equation:
. . $\displaystyle \begin{array}{ccccccc}
(6,0)\!: & 36a + 6b + c &=& 0 \\
(\text{}6,0)\!: & 36b  6b + c &=& 0 \\
(0,18)\!: & 0a + 0b + c &=& 18 \end{array}$
Solve the system of equations: .$\displaystyle a = \text{}\frac{1}{2},\;b = 0,\;c = 18$
Hence, the parabola is: .$\displaystyle y \;=\;\text{}\frac{1}{2}x^2 + 18$
The truck is 8 feet wide.
How high is the tunnel at $\displaystyle x = 4$ ?
Code:
18
*
*  *
*  +  *
*  *
  
*   y *
  
  
*+++*
6 4  4 6

At $\displaystyle x = 4\!:\;y \:=\:\text{}\frac{1}{2}(4^2) + 18 \:=\:10$
Therefore, the truck can be 10 feet high (at most).