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: $\displaystyle a = 25,\:b = 20$

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

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

Substitute the point into the left side of the equation:

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

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

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