Parabolic Minimums

Today's question deals with parabolas, lines, and some calculus. It's not very hard, but, as always, it's very interesting.

Two parabolas, y = x²/x + 7 and y = -x²/4 + 3x, are drawn on the same coordinate plane. A vertical line is going through the parabolas. P and Q are the points of intersections with the first and second parabolas, respectively. Find the vertical line and the shortest distance between P and Q.

The first thing to do when you face such a question is to find an equation that represents the quantity you want, this time the distance PQ. Since the two points are on the same vertical line, they both have the same x-coordinate. Let's call it x. That makes the points:
P(x, x²/2 + 7)
Q(x, -x²/4 + 3x)

The distance between them is just the difference in the y-coordinates (because they lie on a vertical line). The distance is:
d = x² + 7 - (-x²/4 + 3x) = x²/2 + 7 + x²/4 - 3x = 3x²/4 - 3x + 7

We want the shortest d possible, and that's achieved when the derivative of d is zero:
d' = 3x/2 - 3
0 = 3x/2 - 3
3 = 3x/2
6 = 3x
x = 2

Now we know the vertical line, x = 2. Plug this value in d and find the shortest distance between P and Q:
d = 3 * 2²/4 - 3 * 2 + 7 = 3 - 6 + 7 = 4

The shortest distance between P and Q is 4.

Have a great week,
Nadav

nadavs

Geometry and Road Design

Today I have a very interesting question involving geometry. Its solution is so simple that it will make you think "How didn't I think about it?!" when you see the answer. Here it is, try to solve yourself first (as I said, it's easy):

Two towns, town 1 and town 2, are located to the south of road A going in east-west direction. No roads are connecting the towns to each other or to road A. The people of both towns decided to build two roads: one from each town to road A, so they could travel between the towns. What is the shortest route to build these roads?

Well, the shortest route is to build these roads from town to town, but the people of the towns want a connection to road A, so let's give them that.

To solve this question, let's shift our thinking a little bit: let's imagine that town 1 moves to the north of road A, but still keeps the same distance from it (if it was 100 miles to the south, now it's 100 miles to the north).

The shortest distance between two points is on the straight line between them, and so is the road between the towns now. Notice that this road intersects road A.

Now move town 1 back to where it was, and create the road from town 1 to where the roads between the towns intersected road A. It's the same distance as when town 1 was at north, so it must be the shortest distance. Problem solved!

This principle is also known in physics in the field of optics and mirrors. Who said math has no use in real life?

Nadav

nadavs