High school students are learning about the basics of solving quadratics and trigonometric ratios, including trigonometric inverses. The eventual goal of their project is to be able to show a reasonable firing solution, given in initial angle $\theta$ and initial velocity $v_0$. Projectile motion is given by $$y=\left(\tan{\theta}\right)x-\left(\frac{g}{2v_0^2\cos^2{\theta}}\right)x^2$$ where $x$ and $y$ are horizontal and […]

Consider an arbitrary $n$- dimensional hypercube: If we select $n – 1$ corners of that hypercube and highlight all $(n – 2)$ dimensional elements that originate from each of the corners is it possible to cover all the $(n – 2)$ dimensional faces of the cube? Also, is it ever possible to cover all the […]

This is a question posed to my brother in Grade 5. What would be the general approach to solve- How many cuboids of dimensions $a*b*c$ are there in a cuboid of dimension $d*e*f$? My brothers approach: Just get the ratio $\frac{d*e*f}{a*b*c}$ and round it off. My approach: Check 1: Each of $d$, $e$, $f$ must […]

My book states the following: Likewise, we can take a circular cone with base radius $r$ and slant height $l$, cut it along the dashed line in Figure 2: and flatten it to form a sector of a circle with radius $l$ and central angle $ \theta = \frac{2\pi r}{l} $. Why does $ \theta […]

I’m supposed to find the Cayley Table of the group of symmetries for a regular pentagon. But to find the Cayley table, I need to be able to figure out the symmetries of the pentagon. I can see 6 symmetries of a pentagon. The identity, 4 rotations, and 1 reflections on the y-axis. There should […]

This question was inspired to me by Lukas Geyer’s recent question. A positive answer to this question would easily entail a positive answer to Lukas’ question also, and a negative answer would probably be informative as well. Let $T=]0,1[^2$ be the open unit square. Let $(S_k)_{k\geq 1}$ be a countable family of open segments in […]

Suppose $A_1,A_2,A_3,\ldots,A_n$ are distinct points in the plane. A closed path on those points is formed from a permutation of the points, with line segments drawn between successive points (and with a segment connecting the last point to the first, to make the path “closed”). For the six points pictured below on the left, the […]

Suppose I have an some plane and an identity mapping on the points of the plane. I see that the identity can be expressed as a product of an even number of reflections, since any reflection has itself as its own inverse. But why is it impossible to ever express the identity as the product […]

Say you have this type of cubic Bézier curve: The 4 control points A,B,C,D have restrictions: A & B have the same Y-axis coordinate C & D have the same Y-axis coordinate B & C have the same x-axis coordinate My question is: How do you express the X-axis coordinate of the inflection point of […]

The title said it, I want to prove that $$ d(x+u, y + v) \le d(x, y) + d(u,v) $$ for every metric $d$. If the metric is induced by a norm, i.e. $d(x,y) := ||x-y||$, then this is easy. \begin{align*} d(x+u, y+v) & = ||x+u – (y+v)|| \\ & = ||x-y + u – […]

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Applications of the Mean Value Theorem
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$f:\mathbb{Q} \rightarrow \mathbb{R}$, with conditions on $f$
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