From 01c7750a88488b6f8876698363668cd6d4fa01d6 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Anton=20Luka=20=C5=A0ijanec?= Date: Wed, 5 Jan 2022 00:48:35 +0100 Subject: dn --- mat/domace_naloge/1.tex | 103 ++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 103 insertions(+) create mode 100644 mat/domace_naloge/1.tex diff --git a/mat/domace_naloge/1.tex b/mat/domace_naloge/1.tex new file mode 100644 index 0000000..d2a912f --- /dev/null +++ b/mat/domace_naloge/1.tex @@ -0,0 +1,103 @@ +% do-vimlatex-onwrite +\documentclass[]{article} +\usepackage[utf8]{inputenc} +\usepackage{siunitx} +\usepackage[slovene]{babel} +\usepackage[inline]{enumitem} +\usepackage[a4paper, total={7in, 10in}]{geometry} +\usepackage{hologo} +\usepackage[hidelinks,unicode]{hyperref} +\usepackage{datetime} +\usepackage{tkz-euclide} +\usepackage{amssymb} +\usepackage{multicol} +\usepackage{listings} +\usepackage{xcolor} +% \sisetup{output-decimal-marker = {,}, quotient-mode=fraction, per-mode=fraction} % frac način +% \sisetup{output-decimal-marker = {,}, quotient-mode=fraction, per-mode=symbol} % poševnica način +\sisetup{output-decimal-marker = {,}, quotient-mode=fraction} % na -1 način +\settimeformat{hhmmsstime} +\newcommand{\razhroscevanje}{0} +\newcommand{\razhroscevanjeg}{0} % grafično razhroščevanje +\makeatletter +\newcommand{\xslalph}[1]{\expandafter\@xslalph\csname c@#1\endcsname} +\newcommand{\@xslalph}[1]{% + \ifcase#1\or a\or b\or c\or \v{c}\or d\or e\or f\or g\or h\or i% + \or j\or k\or l\or m\or n\or o\or p\or r\or s\or \v{s}% + \or t\or u\or v\or z\or \v{z} + \else\@ctrerr\fi% +} +\AddEnumerateCounter{\xslalph}{\@xslalph}{m} +\makeatother +\definecolor{codegreen}{rgb}{0,0.6,0} +\definecolor{codegray}{rgb}{0.5,0.5,0.5} +\definecolor{codepurple}{rgb}{0.58,0,0.82} +\definecolor{backcolour}{rgb}{0.95,0.95,0.92} +\lstdefinestyle{mystyle}{ + backgroundcolor=\color{backcolour}, + commentstyle=\color{codegreen}, + keywordstyle=\color{magenta}, + numberstyle=\tiny\color{codegray}, + stringstyle=\color{codepurple}, + basicstyle=\ttfamily\footnotesize, + breakatwhitespace=false, + breaklines=true, + captionpos=b, + keepspaces=true, + numbers=left, + numbersep=5pt, + showspaces=false, + showstringspaces=false, + showtabs=false, + tabsize=2 +} +\lstset{style=mystyle} +\title{Domača naloga} +\author{Anton Luka Šijanec, 3. a} +\begin{document} +\maketitle +\begin{abstract} +Rešitev domače naloge, naročene 4. aprila 2021. Navodilo: V učbeniku na strani 49 pri nalogah od prve do devete naredi vsaj štiri primere. +\end{abstract} +% \tableofcontents +\section{Naloge in rešitve} +\begin{enumerate}[label=\textbf{\arabic*.}] + \item Izmeri dolžino in širino klopi ali mize in izračunaj ploščino. Rezultat zaokroži na kvadratni centimeter natančno. + $$\SI{75}{\centi\meter}\cdot\SI{175}{\centi\meter}=\SI{13125}{\centi\meter\squared}$$ + \item Obseg romboida je \SI{70}{\centi\meter}, dolžini sosednjih stranic pa sta v razmerju $4:3$. Kolikšna je ploščina romboida, če je njegov ostri kot velik $\ang{60}$? Rezulat zaokroži na dve decimalni mesti. + $$a = \frac{\SI{70}{\centi\meter}}{7}\cdot3=\SI{40}{\centi\meter} \wedge b=\frac{\SI{70}{\centi\meter}}{7}\cdot3=\SI{30}{\centi\meter} \Rightarrow S = ab\sin\alpha = \SI{40}{\centi\meter}\SI{30}{\centi\meter}\sin\ang{60}=\SI{1039,23}{\centi\meter\squared}$$ + \item Koliko je dolga diagonala pravokotnika, katerega obseg je \SI{22}{\centi\meter}, ploščina pa \SI{30}{\centi\meter\squared}? + $$a+b=\SI{11}{\centi\meter} \wedge ab=\SI{15}{\centi\meter\squared} \wedge \frac{\SI{30}{\centi\meter\squared}}{a}=\SI{11}{\centi\meter}-a=b \Rightarrow $$ + Na tej točki je morebiti že prepozno. Očitno se nikoli nisem naučil osnov matematike. + \begin{lstlisting}[language=Python] +sage: var("a m") +(a, m) +sage: e = 30*m**2/a==11*m-a +sage: solve([e, e2], a, m) +[[a == 5, m == 1], [a == 6, m == 1]]\end{lstlisting} + $$\Rightarrow a = \SI{5}{\centi\meter} \wedge b = \SI{6}{\centi\meter} \Rightarrow d = \sqrt{a^2+b^2}=\sqrt{\SI{25}{\centi\meter\squared}+\SI{36}{\centi\meter\squared}}=\sqrt{\SI{61}{\centi\meter\squared}} = \SI{7,810}{\centi\meter}$$ + \item Diagonali pravokotnika sta dolgi \SI{10}{\centi\meter} in se sekata pod kotom \ang{60}. Izračunaj obseg in ploščino pravokotnika. Rezultat naj bo točen. + $$2\sin\ang{60}\cdot\SI{5}{\centi\meter}=b=\SI[parse-numbers = false]{5\sqrt{3}}{\centi\meter} \wedge 2\cos\ang{60}\cdot\SI{5}{\centi\meter}=a=\SI{5}{\centi\meter} \Rightarrow O = \SI[parse-numbers = false]{5+5\sqrt{3}}{\centi\meter} \wedge S = \SI[parse-numbers = false]{25\sqrt{3}}{\centi\meter\squared}$$ + \item Dolžini stranic pravokotnika sta v razmerju $4:3$. Razpolovišča stranic pravokotnika določajo štirikotnik. + \begin{enumerate}[label=\xslalph*)] + \item Kolikšen del pravokotnika pokrije štirikotnik? + $$S_n=\frac{ef}{2}\sin\theta \wedge \sin\theta=\sin\ang{90}=1 \wedge e=3 \wedge f=4 \Rightarrow S_n=6 \rightarrow \frac{6}{S_z=a_zb_z=12}=\frac{1}{2}$$ + \item Kolikšno je razmerje med obsegoma pravokotnika in štirikotnika? + $$a_n = \sqrt{\frac{4}{2}^2+\frac{3}{2}^2}=\sqrt{6,25}=2,5 \wedge \frac{4\cdot2,5=10}{2\cdot\left(4+3\right)=14}=\frac{5}{7}$$ + \end{enumerate} + \item Ploščina romba, katerega ena izmed diagonal je dolga \SI{36}{\centi\meter}, je \SI{432}{\centi\meter\squared}. Koliko sta dolgi stranica in višina romba? + $$a=\sqrt{\left(\frac{\frac{2\cdot\SI{432}{\centi\meter\squared}}{\SI{36}{\centi\meter}}}{2}\right)^2+\left(\frac{\SI{36}{\centi\meter}}{2}\right)^2}=\SI[parse-numbers = false]{\sqrt{468}}{\centi\meter}$$ + $$v_a=a\sin\alpha \wedge \alpha=2\cdot\left(\ang{90}-\arccos\frac{\SI{18}{\centi\meter}}{a}\right)=\ang{112,6} \Rightarrow o=\SI{19,969}{\centi\meter}$$ + \item Ploščina romboida je \SI{108}{\centi\meter\squared}. Višina na daljšo stranico je dolga \SI{6}{\centi\meter}. Koliko sta dolgi stranici romboida, če sta njuni dolžini v razmerju $2:3$? + $$a=\frac{\SI{180}{\centi\meter\squared}}{\SI{6}{\centi\meter}}=\SI{30}{\centi\meter} \Rightarrow b=\SI{20}{\centi\meter}$$ +\end{enumerate} +\section{Zaključek} +\hologo{LaTeX} izvorna koda dokumenta je objavljena na \url{https://git.sijanec.eu/sijanec/sola-gimb-3}. Za izdelavo dokumenta je potreben \texttt{TeXLive 2020}. +\if\razhroscevanje1 +\vfill +\section*{Razhroščevalne informacije} +Konec generiranja dokumenta \today\ ob \currenttime. + +Dokument se je generiral R0qK1KR2 \SI{}{\second}. % aaasecgeninsaaa +\fi +\end{document} -- cgit v1.2.3