## 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…

## Cyclopentadecane the Simplest Catenane Interlocking Ring Structure?

Notice the links in a gold chain, how they are not connected, but are a series of interlocked rings. There is a word that describes such an unconnected chain: concatenate. From this word, we can gain an understanding of what chemists call a catenane. Now we are probably already familiar with typical multi-ring compounds, such as naphthalene. But naphthalene consists of two hexagonal rings sharing two carbon atoms, and joined together by them. Hence, naphthalene is not a 12-carbon structure, but a 10-carbon structure, C₁₀H₈. Notice the simple illustration of one ring linking to another. This looks simple to achieve, but it is not so easy! We will not discuss the chemistry involved in preparing a catenane, but we will discuss some of the issues. Why Not Simple Ring Closure?…

## 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…

## Is Cyclopropenethione Aromatic, As Is Its Oxygen Analog?

As small as it is, and even with a heteroatom in its make up, cyclopropenethione is aromatic, in the same way cyclopropenone is aromatic. Its aromaticity is not due to a theoretically electrically neutral structure, as in Figure 1, but to its "alternative" zwitterionic structure, shown in Figure 2. Aromatic Characteristics Hückel descriptors fall short of aromaticity if cyclopropenethione exhibited only the structure drawn in Figure 1. In addition to a closed and flat ring, aromaticity requires a 4n+2 number of π-electrons (pi), where n is usually a small integer. In the un-ionized form of Figure 1, each carbon atom has a π-electron, for a total of 3 electrons. This is because every double bond consists of one π-electron plus one σ-electron (sigma) per constituent atom. Widening Perspectives: Cyclopropenethione The…

## Interior Angles of Polygons Starting with a Triangle

Regular polygons are closed curves with a set number of sides and interior angles, each of identical value. How can one calculate the interior angles of polygons? An equilateral triangle has three equal sides and three equal interior angles. Each angle is 60° or π/3 radians, adding up to 180° or π radians total. A square has four equal sides and four equal interior angles. Each interior angle is 90° or π/2 radians, adding up to 360° or 2π radians total. Interior Angles of Polygons A short chart of additional polygons with their interior angles provides the following data: Pentagon 5 interior angles of 108° (540° or 3π radians total) Hexagon 6 interior angles of 120° (720° or 4π radians total) Heptagon 7 interior angles of about 128.57° (900° or 5π radians total)…

## Thumbnail Mystery: Mike Rubino – Music to Die For

Thumbnail Mystery: No one truly liked Mike Rubino. Oh, he had his circle of “friends” and he believed in them. But they all had their reasons to be nice to him. You’d think he’d realize that, but he didn’t. Mike had redeeming qualities. It’s just they were overpowered by the “other” kind. “Hey George, c’mere.” When Mike called, you came. “What? What do you want?” The ‘party of the second part’ was George “Big Nose” Hamlin. One finger or the other was always up-it. George was the closest thing to an actual friend Mike had. Mike viewed George as his “right-hand man.” Uh, yeah. Let's not go there... The catch is, this wore on George’s nerves—George, get this—George, get that. In a low voice Mike said, “D’ja hear from Eddy?”…

## The Algebra Distributive Property – A Simple Introduction

The algebra distributive property lets you multiply a sum by multiplying each part separately and then adding those amounts together. These words are bound to confuse the reader, so let’s consider an example that will demonstrate what we mean. The Example We want to multiply 4x7. Let’s write it as (4)(7). Then, (4)(7) = 28 Now let’s replace 4 with its equivalent, 3+1. And let’s replace 7 with its equivalent, 5+2. Then, (3+1)(5+2) = 28 This seems to be a pretty strange way to write 4x7, doesn’t it? Yet in mathematics – in algebra-style notation – it is just as correct as 4x7. In this form, we can hopefully explain in an understandable way, how the algebra distributive property works. Refer to the diagram to see how we can do…

## Constants and Variables: A Simple Introduction to Algebra

Please bear with me on this article. You see, I am a chemist, not a mathematician. Yet, as an individual who struggled with the concepts behind algebra (yet I grasped it soon enough to ace it), I can understand how others – intelligent individuals – can find algebra disconcerting. What are constants and variables? Two Basic Participants - Constants and Variables There are two primary participants in algebra – variables (which change) and constants (which do not change). Constants are specific numbers that never change. 27 is always 27. 43-1/4 is always 43-1/4. It never changes; it is constant. So let’s consider your age. Your age changes! This year you may be 16. Next year, you will be 17. Age is variable. Let’s write an equation. Your First Algebra Equation…

## A Negative Times a Negative Makes a Positive Number

In high school we were introduced to negative numbers. Why high school? Why not earlier? Because we cannot picture in our minds what a negative number is. We know what positive numbers are. For instance, if we have three apples and someone gives us four more apples, we know we now have seven apples. And as to multiplication, if we have three groups of four three apples each, we know we have 12 apples. But can you visualize what a negative apple might be? How can you demonstrate negative times negative makes positive? Pure Numbers Forget units for the moment. We will concentrate on pure numbers. In the above example, the four groups of three apples becomes simply 4(3) = 12. Suppose, instead of 4(3) we make on of the…

## XY-Coordinate System Symmetry with Examples

[caption id="attachment_28582" align="alignright" width="480"] Image Department of Energy[/caption]In high school mathematics, the topic of symmetry is bound to arise. Especially is this so in analytic geometry. For curve C, what is its XY coordinate system symmetry? How is it symmetric about the y-axis? The x-axis? The origin? The line y = x? The line y = -x? Symmetric about some point other than the origin? Symmetry About the Y-Axis Symmetry about the y-axis means that if there is a curve that lies to the right of the y-axis, there is an identical copy of it to the left of the y-axis. That is, it is symmetrical if each x value can be replaced with –x. Thus, the parabola y = 1/2x2 is symmetric with regard to the y-axis. For every…