# Math Help - Conic Section Application Problems

1. ## Conic Section Application Problems

Problem -
"A semielliptic archway has a height of 20 feet and a width of 50 feet. Can a truck 14 feet high and 10 feet wide drive under the archway without going into the other lane."

I understand that it pertains to an ellipse, but could it essentially be an upside-down hyperbola? Either way, I'm not sure where to actually start. Hopefully somebody can help me, and thank you.

2. Hello, HawthorneKitty!

A semielliptic archway has a height of 20 feet and a width of 50 feet. Can a truck
14 feet high and 10 feet wide drive under the archway without going into the other lane?
Code:
                20|
* * * * *
*       |       *  P
*         * - - - - o(10,14)
*          |         |*
|14       |
*           |    10   | *
* - - - - - * - - - - * *
:    25     :    25     :

We have an ellipse with: $a = 25,\:b = 20$

Its equation is: . $\frac{x^2}{25^2} + \frac{y^2}{20^2} \:=\:1$

Question: Is the upper corner of the truck $P(10,14)$ inside the ellipse?

Substitute the point into the left side of the equation:

. . $\frac{10^2}{25^2} + \frac{14^2}{20^2} \:=\:0.65\:<\:1$

Since the value is less than 1, point $P$ is inside the ellipse.

Yes, the truck can drive through without going into the other lane.