High school geometry questions and answers

Recent questions in High school geometry
Paul Duran 2022-05-09 Answered

Show that $f\left({x}_{1}^{\ast },...{x}_{n}^{\ast }\right)=max\left\{f\left({x}_{1},...,{x}_{n}\right):\left({x}_{1},...,{x}_{n}\right)\in \mathrm{\Omega }\right\}$ if and only if $-f\left({x}_{1}^{\ast },...{x}_{n}^{\ast }\right)=min\left\{-f\left({x}_{1},...,{x}_{n}\right):\left({x}_{1},...,{x}_{n}\right)\in \mathrm{\Omega }\right\}$I am not exactly sure how to approach this problem -- it is very general, so I can't assume anything about the shape of $f$. It seems obvious that flipping the max problem with a negative turns it into a min problem. Thoughts?

Micah Haynes 2022-05-09 Answered

Let $n$ points be placed uniformly at random on the boundary of a circle of circumference 1.These $n$ points divide the circle into $n$ arcs.Let ${Z}_{i}$ for $1\le i\le n$ be the length of these arcs in some arbitrary order, and let $X$ be the number of ${Z}_{i}$ that are at least $\frac{1}{n}$.What is $E\left[X\right]$ and $Var\left[X\right]$?Any hints will be appreciated. Thanks..(By the way this problem is exercise 8.12 from the book 'Probability and Computing' by Mitzenmacher and Upfal)

Alexis Meyer 2022-05-09 Answered

The length of an arc of a circle is 12 cm. The corresponding sector area is 108 cm${}^{2}$. Find the radius of the circle.I have not attempted this question and do not understand how to solve this.

Carina Valenzuela 2022-05-08 Answered

Let $X\in \left\{0,1{\right\}}^{d}$ be a Boolean vector and $Y,Z\in \left\{0,1\right\}$ are Boolean variables. Assume that there is a joint distribution $\mathcal{D}$ over $Y,Z$ and we'd like to find a joint distribution ${\mathcal{D}}^{\prime }$ over $X,Y,Z$ such that:1. The marginal of ${\mathcal{D}}^{\prime }$ on $Y$,$Z$ equals $\mathcal{D}$.2. $X$ are independent of $Z$ under ${\mathcal{D}}^{\prime }$, i.e., $I\left(X;Z\right)=0$.3. $I\left(X;Y\right)$ is maximized,where $I\left(\cdot ;\cdot \right)$ denotes the mutual information. For now I don't even know what is a nontrivial upper bound of $I\left(X;Y\right)$ given that $I\left(X;Z\right)=0$? Furthermore, is it possible we can know the optimal distribution ${\mathcal{D}}^{\prime }$ that achieves the upper bound?My conjecture is that the upper bound of $I\left(X;Y\right)$ should have something to do with the correlation (coupling?) between $Y$ and $Z$, so ideally it should contain something related to that.

Jay Barrett 2022-05-08 Answered

Given a right circular cone with the line of symmetry along $x=0$, and the base along $y=0$, how can I find the maximum volume paraboloid (parabola revolved around the y-axis) inscribed within the cone? Maximising the volume of the paraboloid relative to the volume of the right circular cone. In 2-D, the parabola has 2 points of tangency to the triangle, one of each side of the line of symmetry. I have tried using the disk method to find the volume of the cone, and the parabola, both with arbitrary equations such as $y=b-ax$, and $y=c-d{x}^{2}$, but I end up with a massive equation for several variables, instead of a simple percentage answer. Any help is appreciated! Thanks in advance.

Matthew Hubbard 2022-05-08 Answered

How did Pythagoras argue for the Pythagorean theorem?There are lots of proofs for the Pythagorean theorem. But, how did Pyhtagoras argue for the theorem himself?

Logan Lamb 2022-05-08 Answered