Quadratic EquationsMore Applications

It turns out that there are several different types of paths an object can take when gravity, and these are called conic sections.

Conic Sections

The light rays emitted from a flashlight form a cylinder. If you shine in onto a flat surface, you can see the different shapes that can be created by slicing though a cone:

If you point the flashlight directly onto the ground you get a circleellipseoval. If you tilt it slightly, the circle becomes an ellipse. If you continue rotating the flashlight, the ellipse becomes larger and larger, and eventually turns into a U-shape that continues forever: this is a Parabola. And if you keep going, the parabola will become a hyperbola, a different kind of shape that you'll learn more about later.

The parabola is produced by slicing the cone exactly parallel to its edge. You can prove that its shape matches the one created by graphing a quadratic equation – but that is still a bit too difficult.

We've already seen that parabolae describe the path of objects thrown into the air. All planets in our solar system move on elliptical orbits around the sun, and the XXXXX.

When NASA was launching the first rockets into space, mathematicians like [Kathrine Johnson] had to calculate the exact paths taken by the spacecrafts. The first launches were simple parabolas that only took a few minutes. Later launches then changed into

https://hackaday.com/2018/02/28/katherine-johnson-computer-to-the-stars/

diagram

Calculating the transition between parabolic and elliptical orbits was an incredibly difficult task – as explained here in the move "Hidden Figures":

Quadratic functions and equations appear not just in the motion of projectiles, but have many other applications in science, engineering, economics and nature. Let's have a look at a few more applications.

Mirrors and Reflections

Draw two points on a sheet of paper, and make a fold that brings one of the points directly on top of the other. You can think of this fold as the "line of reflection" between A and B. Every point on the line has the same distance between A and B.

We can say, "the set of points equidistant from A and B are precisely the points on the perpendicular bisector of A and B."

Take a blank piece of paper and draw on it a straight line and a point P not on that line. Use a thick marker to make the point conspicuous.

Holding the paper up to the light, fold the paper in such a way that the point lands somewhere on the line. Make a sharp crease.

Unfold the paper and then make a second that takes the point P to a different place on the line. Make a second sharp crease.

Repeat this action at least FIFTY more times, making another fifty creases in the paper given by taking the point P to different locations on the line.

The 52 creases you have outline the shape of an interesting curve.

With a pencil draw along the creases to outline the curve you see.

When completing this activity, you should see a U-shaped curve. The question, of course, is this curve the shape of a quadratic/parabola?

The previous folding activity asked as to measure distances between a point and line. Recall that we measure the distance of a point from a line via the length of a perpendicular line segment from the point to the line:

Suppose we are given a line L and a point F not on that line. Then the set of all points that are equidistant from the point and from the line (see the diagram below) give a U-shaped curve.

The point F is called the focus of this special curve and the line L is called its directrix.

This curve looks U-shaped. Is it quadratic?

Parabolas have the astounding reflection property that all incoming parallel rays of light, perpendicular to the directrix, are directed to the focus of the parabola.

Satellite dishes and reflecting telescopes use dishes with parabolic cross-sections so as to focus parallel rays of light to a fixed point, and conversely, search-light reflectors and automobile headlight reflectors, for example, are parabolic: all rays from a bulb positioned at the focus are reflected parallel to the axis of the parabola.

In a parabolic mirror, incoming light rays are all reflected onto a single point in the center. This curious property can be used when building radio telescopes, where the receiver is placed in the focus, or thermal solar power plants, which focus the suns' rays onto a single point to create a lot of heat.

The actual shape of this telescope is called a Paraboloid – it is basically a three-dimensional version of a parabola.

Even the TV dishes at your home have a paraboloid shape. They focus the incoming radio waves onto the small receiver in its center.

Bridges

In many suspension bridges the main suspension cables form a parabola. One famous example is the Golden Gate Bridge in San Francisco:

Suspension bridges can span especially long distances, and are relatively economical to build.

The suspension cables carry the huge weight of the deck of the bridge, as well as all the traffic travelling across it. This causes large tension and compression forces, and the parabola is the best shape to balance these forces equally.

Catenary Curves

Galileo noticed shape of a rope or a chain hanging between two poles is U-shaped and deduced that it too must be given by a quadratic formula. (We see this with the shape of power lines, the shape of ropes that surround sculptures in art museums, and so on.)

As we saw in part I, Galileo also wondered if the shapes of hanging chains were also the shapes of quadratic curves.

It turns out they are not! These curves are called catenary curves (from the Latin word for “chain”) and are given by a complicated formula. Their mathematics were not properly figured out until 1691, once the subject of calculus was invented.

Conduct some internet research to learn about catenary curves, catenary arches, and their history.

It turns out that Galileo was in error about the shapes of hanging chains: they do not follow a quadratic formula. Trust what your data is telling you!

The shape of the curve given by a hanging chain became known as a catenary curve (from the Latin catena for “chain). It wasn’t until 1691 that mathematicians found a precise formula for this curve. (It is very different from a quadratic formula!)

EXTENDED ACTIVITY: Is the St. Louis Gateway Arch in the shape of a quadratic curve? Find out by making measurements on a photograph of the arch.

The Multiplication Parabola

http://demonstrations.wolfram.com/TheMultiplicationParabola/

More Applications

https://demonstrations.wolfram.com/LiquidInARotatingCylinder/

Maximum and minimum values of quadratics are used in the study of moving objects and in acceleration and volume problems. Business models also include quadratic functions and are used to help forecast profit and loss.

Many equations in physics or economics contain multiplication, and these multiplications often lead to squares. That’s why it should come at no surprise that our world is full of quadratic equations and parabolas.

Quadratic equations are hidden everywhere in our world. Can you think of any other examples?