Hello again, archistrategos214!
3) A waterway whose cross-section is a parabola is being designed for a theme park boat ride.
It has a depth of 12 ft and a width of 6 ft at the surface.
Rectangular cross-sectioned boats are to be used, and they sink to a depth of 4 ft.
What should be the proper width of the boat to facilitate easy passage? Code:
|
* - -*------------+--------------*- - *(3,12)
| | |
* | | 4| *
* | x | x | *
*------------+--------------*
* | * :y
* | * :
- - - - - - - - - - * - - - - - - -+- - -
|
The parabola has the equation: .$\displaystyle y \,=\,ax^2$
The point $\displaystyle (3,12)$ is on the parabola.
. . We have: .$\displaystyle 12 \:=\:a\cdot3^2\quad\Rightarrow\quad a = 2$
The parabola is: .$\displaystyle y \,=\,2x^2$
Since the boats sink to 4 feet, then: .$\displaystyle 12 - y = 4$
. . Then: .$\displaystyle 12 - 2x^2\:=\:4\quad\Rightarrow\quad x \,=\,2$
Therefore, the boatrs should be $\displaystyle 2x = 4$ feet wide at most.