Conic Section Application Problems

• May 31st 2009, 05:57 PM
HawthorneKitty
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.
• May 31st 2009, 07:21 PM
Soroban
Hello, HawthorneKitty!

Quote:

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.