## Using Data from Graphs: Interpolation Vs. Extrapolation

[caption id="attachment_28677" align="alignright" width="480"] Fig. 1. Plot of cook time vs. temperature[/caption]A familiar technique used when collecting data is to graph the results. For instance, suppose you want to see how quickly the internal temperature of a roast of pork rises in a 250° F oven. Nine internal meat temperature measurements are taken over a period of an hour-and-a-half, or 180 minutes. Collecting the Data After 20 minutes, the internal temperature of our pork roast is 60° F. Twenty more minutes yields 95° F. At 60 minutes, the temperature is 118° F, whereas the temperature is 139° F after 80 minutes. At 100 minutes, we read 148° F. When two hours have passed, we obtain 156° F. 140 minutes of cooking puts us at 163° F, while 160 minutes gives…

## How Much Does a Fly Hitting a Train Slow It Down?

[caption id="attachment_27068" align="alignright" width="480"] The Scenario: a fly hitting a train.[/caption]Say you have a train, a locomotive traveling along a straight railroad track. The atmosphere is absolutely free of every conceivable impediment except for one lone housefly. Unfortunately (for the fly), it and the train collide head-on. You have a fly hitting a train. Clearly, the fly will turn into a smear. How much does that fly slow the train down that it has hit? The train will suffer no apparent change in speed, yet it does slow down, even if there is no obvious change. How much does it slow down? Fly Hitting a Train: Specific Numbers We will attempt to use realistic numbers. We choose the following: Locomotive: 4,000 tons [a light train] Fly: 14 milligrams Locomotive Speed:…

## Is Gravity in Atoms Significant? No. Prove It!

[caption id="attachment_23683" align="alignright" width="480"] Is there gravity in the atom?[/caption] Theories of the microscopic never seem to include reference to gravity in the atom. Should they? What do you think? Numbers don’t lie: The reality is, gravity inside the atom is pretty insignificant. Let’s look at this in terms of scale, and then examine the equations for determining gravitational pull. Atoms and our Scale of Reference It is the human tendency to draw conclusions – with reference to the extremely large and the extremely small – on the basis of what we experience in our scale of reference. In fact, much good science has been realized using such assumptions. But only much good science – by no means all. In fact, many of the most incredible discoveries have not been…

## Mathematics Intrinsic Value, or Simply a Tool?

It is a sunny day; you want to take a walk. We here define walking as the taking of repeated steps in any given direction. Our walk will take us into the field of mathematics. An Unusual Walk Imagine a stretch of sidewalk that is precisely 6 feet long. An initial 3 foot step is taken. Every step after that must be 1/2 the distance of the previous step. So our second step is 1/2 the length of the first step. The third step is 1/2 the length of the second step, and so on. Questions: 1. Theoretically, how long would it take you to reach the end of the 6′stretch? 2. How far would you have traveled, after each step? Answers: 1. You would, again theoretically, never make it…

## Simplifying Mathematics: Introducing Vectors and Vector Addition

[caption id="attachment_23153" align="alignright" width="480"] Magnetic vector force field[/caption] We all know how to add, subtract, multiply and divide ordinary numbers, even if basic units are attached to them such as gallons, apples, feet, tons, and so forth. 4+3 = 7 5 lbs x 3.2 = 16 lbs These two examples illustrate pure numbers in the first instance, and simple quantities in the second instance. Introducing Vectors: What About Direction? What if we toss in direction? Imagine a huge square, 5 miles on a side. We have to travel along the perimeter to travel from Point A to Point B, and then on to our destination, Point C (see the image). We thus travel 10 miles to reach Point C. If we could travel "as the crow flies", we would only…

## How Far Can Perfect Eyes See? An Ideal Earthly Scenario

[caption id="attachment_26540" align="alignright" width="480"] Image by Meg Learner[/caption]I once met a fellow in Virginia who said, 'The human eye is an amazing thing. Why, if there were no mountains in-between, we could see California!' Of course, he was referring to perfect eyes... Of course, that is just plain nuts. Nevertheless, it raised the question, "Just how far away could a person see an object if nothing interfered? Let's consider the answer to that question. Conditions First, we need to set conditions or ground rules. Earth is sufficiently round to call it a sphere, so we treat it as such. In fact, we assume it is perfectly smooth even to an ant. Further, we assume the atmosphere is perfectly clear, and the observer has perfect eyes. When we look out toward…

## Deriving Basic (Circular) Trigonometric Functions

[caption id="attachment_18674" align="alignright" width="401"] Figure 1[/caption] Draw two intersecting lines in space, illustrated below. The mathematician will not want to leave this simple drawing without completely pointing out its features and labeling those features. We do so to begin our understanding of basic (circular) trigonometric functions. [caption id="attachment_18675" align="alignleft" width="387"] Figure 2[/caption] We label the point of intersection of course – P will do. But the intersection produces what looks like slices in a pie. The size of those slices of pie were determined by how the two lines intersected, how "wide apart" the lines are. We label these as angles α (alpha) and β (beta). Superimposing a Circle [caption id="attachment_18676" align="alignright" width="387"] Figure 3[/caption] The title of this paper is understanding basic circular trigonometric functions. So at this stage,…

## Simple Algebra II Graph Symmetries Discussion and Examples

[caption id="attachment_17854" align="alignright" width="480"] Typical functions in two variables.[/caption] College preparatory classes in high school often include Algebra and Algebra II. Perhaps the most memorable aspect of Algebra II is the two-dimensional (2D) graphing of mathematical functions in two variables. This is typically introduced beginning with the Cartesian coordinate system. The generic function is written y = f (x). This reads y equals a function of x. See the illustration for some examples of functions. Cartesian Coordinate System In the Cartesian system, two variables, often x and y, are assigned their own line, one horizontal (x), one vertical (y). The intersection between the two axes is called the origin, and is assigned the value (0, 0). The value of x is the value written on the left in the brackets;…

## Sphere Reciprocal? Not Inside Out, But Equation Inverse?

One way of mathematically representing a very simple sphere in 3D space is, r2 = x2 + y2 + z2 where r equals a radius of the sphere. Solving in terms of x, y, and z, we get, x = √(r2 — y2 — z2) y = √(r2 — x2 — z2) z = √(r2 — x2 — y2) A Sphere Reciprocal Now a sphere may be the most aesthetically pleasing of the simple geometric curves. So it is natural to wonder, concerning a sphere, what if…? So what if we convert the equation into an equation for a sphere reciprocal? No, not turn the sphere inside out. Rather an inverse of the equation of a sphere? What is the graph of, r2 = 1/( x2 + y2 + z2)…