The SAT Question Everyone Got Wrong

What people said

Cites Aadya Chaudhary · 2023-12 · Mathematics Stack Exchange · on “Winding a wire of definite width around a right cylinder”

The original question is presented like this: A copper wire, $3~mm$ in diameter is wound about a cylinder whose length is $12~cm$ and diameter $10~cm$ , so as to cover the curved surface area of the cylinder. Find the length of the wire. Answer: $400~\pi$ I am a highschool student and with my teacher's help and my own start I came to the bookish answer. The process was like you count the no. of loops to be placed around the cylinder and then multipy the length of wire in 1 loop and no. of loops to get the final length of wire. Everthing was good, it came out to be 40 loops and now the problem starts. According to my teacher and my own second start (At first I thought the same that I am think

Cites Abezhiko · 2024-02 · Mathematics Stack Exchange · on “Derivation of an epitrochoid”

I'll detail the approach in J.G.'s comment. Let's work in the complex plane in order to deal with both coordinates at the same time, as suggested by J.G. From now on, I'll refer to the linked animation in the same comment, with the same initial situation. Let $R$ and $r$ be the radii of the static big circle (centered at the origin) and the rotating little one respectively. Let $z = x + iy$ be a point of the epitrochoid and $\theta$ the angle between the $x$ -axis and the center of the rotating circle, so that $\theta = 0$ initially. Finally, let's denote the distance between the point $z$ and the center of the little by $d$ . In consequence, the position of the center of the little circle i

Cites dan_fulea · 2024-04 · Mathematics Stack Exchange · on “Find all square roots of this matrix using some practical method”

Let $B$ be the matrix under the radical, $$ B = \begin{bmatrix} 1 & 3 & -3 \\ 0 & 4 & 5 \\ 0 & 0 & 9 \end{bmatrix}\ . $$ This answer solves the following problem: Find all $3\times 3$ matrices $A$ so that: $$A^2 =B\ .$$ Short solution: The matrix $A_0=\left[\begin{smallmatrix}1 & 1 & -1 \\ & 2 & 1\\& & 3\end{smallmatrix}\right]$ is one solution, found by searching it in upper triangular form. Let $D$ be diagonal with entries $1,2,3$ , so the Jordan decomposition of $A_0$ is of the shape $A_0=SDT$ , with $ST=I$ . Let $A$ be any solution of $A^2=B$ . Associate $X=TAS$ . Then $X^2 =TA^2S=TBS=TA_0^2S=(TA_0S)^2=D^2$ . So $X^2-k^2I$ has rank two for $k^2\in\{1^2,2^2,3^2\}$ , implying that the eige

Cites dan_fulea · 2024-04 · Mathematics Stack Exchange · on “Find all square roots of this matrix using some practical method”

@PatrickStevens Before some lines will be removed, please give a short answer to the question "*what is $\sqrt {D^2}$ where $D$ is the diagonal matrix with entries $1,2,3$?*". To have something to compare with, ethically, mathematically, didactically, and in order to have an exact problem at the base of the question, consider: [this video] (The SAT question everyone got wrong) Sometimes, writing an answer that covers the undefined parts of the question in all of its range is harder than the intention. This must become an issue.

Cites Venkata Karthik Bandaru · 2025-03 · Mathematics Stack Exchange · on “Circle revolutions rolling around another circle”

Link to Veritasium video: Link [this video] Wrong Picture (!) : Note that here the bigger radius has rotated by an angle $${ \frac{{\color{blue}{\text{length of blue curve}}}}{3r} = \frac{r \Delta \theta}{3 r} = \frac{\Delta \theta}{3} . }$$ Hence the instantaneous velocity of ${ {\color{red}{P}} }$ at ${ t = 0 }$ is $${ {\begin{aligned} &\, v _{{\color{red}{P}}} (0) \\ = &\, \frac{d}{dt} (\vec{O O ^{’}} + \vec{O ^{’} P} ) \Big \vert _{t = 0} \\ = &\, \frac{d}{dt} \vec{O O ^{’}} \Big \vert _{t = 0} + \frac{d}{dt} \vec{O ^{’} P} \Big \vert _{t = 0} \\ = &\, \lim _{\Delta t \to 0} \frac{(3r + r) \frac{\Delta \theta}{3}}{\Delta t} \, \hat{x} + \lim _{\Delta t \to 0} \frac{r \Delta \theta}{\Delta t }