# Tutor profile: Aaron P.

## Questions

### Subject: Optical Science

Let's say I have a lens with a known refractive index (n) and I know the object location and the lens's focal length. I know the lens is plano-convex, but I don't know what the radius of curvature is. How can I find the radius of curvature?

This is best answered by looking at the Lensmaker's equation: $$1/f = (n-1)*(1/R_1 - 1/R_2 + [(n-1)*d)]/ [n*R_1*R_2]) $$ Given that the lens is plano-convex, $$R_1 = Infinity$$, since an infinite radius of curvature gives us a flat surface. This means the first term, $$1/R_1$$ is 0, since any number divided by infinity is 0. This also means the third term is 0, since the denominator ($$n*R_1*R_2$$) is also infinite. This leaves us with a much simpler equation, $$1/f = (n-1) * -(1/R_2)$$ But as you mentioned, we know the refractive index ($$n$$), and we know the focal length, $$f$$, so we can easily solve for $$R_2$$.

### Subject: Data Engineering

Let's talk trees. Let's say I want to plot every tree in a forest on an informative graph. A single tree is a function of, say, six parameters: Height, trunk width, age, latitude, and longitude, and altitude. Is it possible to show someone all this information at once?

I would say Yes. Since you have 6 parameters, the problem boils down to generating a 2-dimensional image to represent 6-dimensional data, and this can be done with a 3-dimensional plot. The human brain can easily interpret 3-D images even through they are viewed on a 2D plane, like a computer or picture. So we'll select parameters for our first three parameters, which we can represent on X, Y, and Z axes. let's say Latitude is X, Longitude is Y, and altitude is Z. This will give us an intuitive correlation between the tree's location in the forest and its location in the 3D plot. A single point at the intersection of each X,Y,Z coordinate tells us that a tree exists at some specific coordinate, but nothing else about it. To represent the trunk diameter, we can use the width of the dot at the location -- the wider the trunk, the wider the point. We can represent age by the _color_ of the point. For instance, a black circle could be an older tree. Finally, the height can be indicated by the _shape_ of the point. For instance, young trees could be a small, green circle, while a larger, black square would represent an older tree!

### Subject: Physical Science

When I stick my hand in a fish tank full of water and view it from the side, it seems broken. Why?

This is one of those fantastic examples of when Physics seems to "break". In reality, your hand does not separate suddenly from your arm -- that would hurt! What you are seeing is based on how your eye sees light rays traveling from different parts of your hand -- an effect we call "refraction". Before it enters the tank, your hand is in air -- to quantify this a bit, let's say air has a "thickness" of 1. When your hand enters the water, light needs to pass through the water to reach your eyes. The water, though, has a different "thickness" to light -- roughly 1.4. Light, though, is lazy, and it likes to go at the same speed all the time. So when light enters the water, it needs to cover the same distance in the same amount of time! Since water is "thicker" to light than air, it needs to change the path it travels. This causes the light rays to change the angle at which they're traveling through the water! Coincidentally, these are the light rays that reach your eyes. The "broken" appearance is just due to light traveling in a different direction once it crosses the air-water boundary!